Octave 3.8, jcobi/3

Percentage Accurate: 94.3% → 99.8%

Time: 20.6s

Alternatives: 20

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	return (((1.0 + alpha) / t_0) / t_0) * ((1.0 + beta) / (3.0 + (alpha + 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
    real(8) :: t_0
    t_0 = alpha + (2.0d0 + beta)
    code = (((1.0d0 + alpha) / t_0) / t_0) * ((1.0d0 + beta) / (3.0d0 + (alpha + beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 86.7%

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

   3. +-commutative 86.7%

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

   4. associate-+r+ 86.7%

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

   5. +-commutative 86.7%

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

   6. associate-+r+ 86.7%

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

   7. associate-+r+ 86.7%

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

   8. distribute-rgt1-in 86.7%

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

   9. +-commutative 86.7%

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

   10. *-commutative 86.7%

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

   11. distribute-rgt1-in 86.7%

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

   12. +-commutative 86.7%

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

   13. times-frac 96.7%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

   2. expm1-udef 69.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

   3. *-commutative 69.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

   4. +-commutative 69.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

5. Applied egg-rr 69.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

   2. expm1-log1p 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r/ 96.8%

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

   5. times-frac 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \alpha}{t_0} \cdot \frac{1 + \beta}{t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 5.4e+18)
     (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 beta) (* t_0 (+ beta (+ alpha 3.0)))))
     (*
      (/ (+ 1.0 beta) (+ 3.0 (+ alpha beta)))
      (/ (/ (+ 1.0 alpha) beta) t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5.4e+18) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (beta + (alpha + 3.0))));
	} else {
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / t_0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 5.4d+18) then
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + beta) / (t_0 * (beta + (alpha + 3.0d0))))
    else
        tmp = ((1.0d0 + beta) / (3.0d0 + (alpha + beta))) * (((1.0d0 + alpha) / beta) / t_0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5.4e+18) {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (beta + (alpha + 3.0))));
	} else {
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / t_0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 5.4e+18:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (beta + (alpha + 3.0))))
	else:
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / t_0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 5.4e+18)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(t_0 * Float64(beta + Float64(alpha + 3.0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(3.0 + Float64(alpha + beta))) * Float64(Float64(Float64(1.0 + alpha) / beta) / t_0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 5.4e+18)
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + beta) / (t_0 * (beta + (alpha + 3.0))));
	else
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.4e+18], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.4e18

   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/ 98.6%

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

      2. associate-/r* 94.6%

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

      3. +-commutative 94.6%

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

      4. associate-+r+ 94.6%

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

      5. +-commutative 94.6%

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

      6. associate-+r+ 94.6%

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

      7. associate-+r+ 94.6%

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

      8. distribute-rgt1-in 94.6%

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

      9. +-commutative 94.6%

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

      10. *-commutative 94.6%

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

      11. distribute-rgt1-in 94.6%

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

      12. +-commutative 94.6%

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

      13. times-frac 98.6%

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

   3. Simplified 98.6%

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

   if 5.4e18 < beta

   1. Initial program 85.4%

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

      1. associate-/l/ 78.8%

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

      2. associate-/r* 67.8%

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

      3. +-commutative 67.8%

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

      4. associate-+r+ 67.8%

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

      5. +-commutative 67.8%

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

      6. associate-+r+ 67.8%

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

      7. associate-+r+ 67.8%

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

      8. distribute-rgt1-in 67.8%

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

      9. +-commutative 67.8%

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

      10. *-commutative 67.8%

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

      11. distribute-rgt1-in 67.8%

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

      12. +-commutative 67.8%

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

      13. times-frac 92.2%

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

   3. Simplified 92.2%

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

      1. expm1-log1p-u 92.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 55.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 92.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 92.2%

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

      3. *-commutative 92.2%

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

      4. associate-*r/ 92.2%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in beta around inf 83.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}\\ \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{\alpha + 2}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\beta} - \frac{\alpha + 2}{\beta} \cdot \frac{2}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 alpha) (+ alpha (+ 2.0 beta)))))
   (if (<= beta 8.5)
     (* t_0 (/ (/ 1.0 (+ alpha 2.0)) (+ alpha 3.0)))
     (* t_0 (- (/ 1.0 beta) (* (/ (+ alpha 2.0) beta) (/ 2.0 beta)))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	double tmp;
	if (beta <= 8.5) {
		tmp = t_0 * ((1.0 / (alpha + 2.0)) / (alpha + 3.0));
	} else {
		tmp = t_0 * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + alpha) / (alpha + (2.0d0 + beta))
    if (beta <= 8.5d0) then
        tmp = t_0 * ((1.0d0 / (alpha + 2.0d0)) / (alpha + 3.0d0))
    else
        tmp = t_0 * ((1.0d0 / beta) - (((alpha + 2.0d0) / beta) * (2.0d0 / beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	double tmp;
	if (beta <= 8.5) {
		tmp = t_0 * ((1.0 / (alpha + 2.0)) / (alpha + 3.0));
	} else {
		tmp = t_0 * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (1.0 + alpha) / (alpha + (2.0 + beta))
	tmp = 0
	if beta <= 8.5:
		tmp = t_0 * ((1.0 / (alpha + 2.0)) / (alpha + 3.0))
	else:
		tmp = t_0 * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(2.0 + beta)))
	tmp = 0.0
	if (beta <= 8.5)
		tmp = Float64(t_0 * Float64(Float64(1.0 / Float64(alpha + 2.0)) / Float64(alpha + 3.0)));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 / beta) - Float64(Float64(Float64(alpha + 2.0) / beta) * Float64(2.0 / beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + alpha) / (alpha + (2.0 + beta));
	tmp = 0.0;
	if (beta <= 8.5)
		tmp = t_0 * ((1.0 / (alpha + 2.0)) / (alpha + 3.0));
	else
		tmp = t_0 * ((1.0 / beta) - (((alpha + 2.0) / beta) * (2.0 / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.5], N[(t$95$0 * N[(N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 / beta), $MachinePrecision] - N[(N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision] * N[(2.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 94.4%

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

      3. +-commutative 94.4%

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

      4. associate-+r+ 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.3%

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

      7. associate-+r+ 94.4%

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

      8. distribute-rgt1-in 94.4%

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

      9. +-commutative 94.4%

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

      10. *-commutative 94.4%

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

      11. distribute-rgt1-in 94.3%

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

      12. +-commutative 94.3%

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

      13. times-frac 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in beta around 0 97.6%

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

      1. associate-/r* 98.9%

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

      2. +-commutative 98.9%

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

   6. Simplified 98.9%

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

   if 8.5 < beta

   1. Initial program 86.6%

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

      1. associate-/l/ 80.6%

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

      2. associate-/r* 70.5%

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

      3. +-commutative 70.5%

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

      4. associate-+r+ 70.5%

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

      5. +-commutative 70.5%

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

      6. associate-+r+ 70.5%

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

      7. associate-+r+ 70.5%

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

      8. distribute-rgt1-in 70.5%

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

      9. +-commutative 70.5%

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

      10. *-commutative 70.5%

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

      11. distribute-rgt1-in 70.5%

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

      12. +-commutative 70.5%

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

      13. times-frac 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. metadata-eval 83.5%

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

      5. distribute-lft-in 83.5%

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

      6. *-commutative 83.5%

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

      7. unpow2 83.5%

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

      8. times-frac 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \frac{\frac{1}{\alpha + 2}}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)} \cdot \left(\frac{1}{\beta} - \frac{\alpha + 2}{\beta} \cdot \frac{2}{\beta}\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 3 + \left(\alpha + \beta\right)\\ t_1 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{t_1}}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{\frac{1 + \alpha}{\beta}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 3.0 (+ alpha beta))) (t_1 (+ alpha (+ 2.0 beta))))
   (if (<= beta 1.12e+32)
     (/ (/ (+ 1.0 (+ alpha beta)) t_1) (* t_0 t_1))
     (* (/ (+ 1.0 beta) t_0) (/ (/ (+ 1.0 alpha) beta) t_1)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.12e+32) {
		tmp = ((1.0 + (alpha + beta)) / t_1) / (t_0 * t_1);
	} else {
		tmp = ((1.0 + beta) / t_0) * (((1.0 + alpha) / beta) / t_1);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (alpha + beta)
    t_1 = alpha + (2.0d0 + beta)
    if (beta <= 1.12d+32) then
        tmp = ((1.0d0 + (alpha + beta)) / t_1) / (t_0 * t_1)
    else
        tmp = ((1.0d0 + beta) / t_0) * (((1.0d0 + alpha) / beta) / t_1)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = 3.0 + (alpha + beta);
	double t_1 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 1.12e+32) {
		tmp = ((1.0 + (alpha + beta)) / t_1) / (t_0 * t_1);
	} else {
		tmp = ((1.0 + beta) / t_0) * (((1.0 + alpha) / beta) / t_1);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = 3.0 + (alpha + beta)
	t_1 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 1.12e+32:
		tmp = ((1.0 + (alpha + beta)) / t_1) / (t_0 * t_1)
	else:
		tmp = ((1.0 + beta) / t_0) * (((1.0 + alpha) / beta) / t_1)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(3.0 + Float64(alpha + beta))
	t_1 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 1.12e+32)
		tmp = Float64(Float64(Float64(1.0 + Float64(alpha + beta)) / t_1) / Float64(t_0 * t_1));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / t_0) * Float64(Float64(Float64(1.0 + alpha) / beta) / t_1));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = 3.0 + (alpha + beta);
	t_1 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 1.12e+32)
		tmp = ((1.0 + (alpha + beta)) / t_1) / (t_0 * t_1);
	else
		tmp = ((1.0 + beta) / t_0) * (((1.0 + alpha) / beta) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.12e+32], N[(N[(N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.12000000000000007e32

   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/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

      8. +-commutative 98.6%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in beta around 0 97.4%

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

   if 1.12000000000000007e32 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 77.3%

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

      2. associate-/r* 66.8%

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

      3. +-commutative 66.8%

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

      4. associate-+r+ 66.8%

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

      5. +-commutative 66.8%

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

      6. associate-+r+ 66.8%

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

      7. associate-+r+ 66.8%

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

      8. distribute-rgt1-in 66.8%

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

      9. +-commutative 66.8%

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

      10. *-commutative 66.8%

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

      11. distribute-rgt1-in 66.8%

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

      12. +-commutative 66.8%

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

      13. times-frac 91.6%

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

   3. Simplified 91.6%

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

      1. expm1-log1p-u 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 58.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-*r/ 91.7%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in beta around inf 86.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta\right)}{\alpha + \left(2 + \beta\right)}}{\left(3 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6e+31)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (*
    (/ (+ 1.0 beta) (+ 3.0 (+ alpha beta)))
    (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+31) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / (alpha + (2.0 + 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 <= 6d+31) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + beta) / (3.0d0 + (alpha + beta))) * (((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta)))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6e+31) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / (alpha + (2.0 + beta)));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 6e+31:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / (alpha + (2.0 + beta)))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6e+31)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(3.0 + Float64(alpha + beta))) * Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta))));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6e+31)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + beta) / (3.0 + (alpha + beta))) * (((1.0 + alpha) / beta) / (alpha + (2.0 + 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, 6e+31], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.99999999999999978e31

   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/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

      8. +-commutative 98.6%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in alpha around 0 80.4%

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

   5. Taylor expanded in alpha around 0 60.8%

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

      1. +-commutative 56.4%

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

   7. Simplified 60.8%

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

   if 5.99999999999999978e31 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 77.3%

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

      2. associate-/r* 66.8%

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

      3. +-commutative 66.8%

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

      4. associate-+r+ 66.8%

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

      5. +-commutative 66.8%

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

      6. associate-+r+ 66.8%

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

      7. associate-+r+ 66.8%

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

      8. distribute-rgt1-in 66.8%

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

      9. +-commutative 66.8%

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

      10. *-commutative 66.8%

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

      11. distribute-rgt1-in 66.8%

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

      12. +-commutative 66.8%

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

      13. times-frac 91.6%

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

   3. Simplified 91.6%

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

      1. expm1-log1p-u 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 58.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-*r/ 91.7%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in beta around inf 86.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \beta}{3 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 6: 98.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9e+31)
   (/ (/ (+ 1.0 beta) (+ 2.0 beta)) (* (+ 2.0 beta) (+ beta 3.0)))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ 2.0 beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9e+31) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + 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 <= 9d+31) then
        tmp = ((1.0d0 + beta) / (2.0d0 + beta)) / ((2.0d0 + beta) * (beta + 3.0d0))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (2.0d0 + beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9e+31) {
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 9e+31:
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9e+31)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(2.0 + beta)) / Float64(Float64(2.0 + beta) * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(2.0 + beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9e+31)
		tmp = ((1.0 + beta) / (2.0 + beta)) / ((2.0 + beta) * (beta + 3.0));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (2.0 + 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, 9e+31], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + beta), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.9999999999999992e31

   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/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

      8. +-commutative 98.6%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in alpha around 0 80.4%

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

   5. Taylor expanded in alpha around 0 60.8%

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

      1. +-commutative 56.4%

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

   7. Simplified 60.8%

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

   if 8.9999999999999992e31 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 77.3%

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

      2. associate-/r* 66.8%

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

      3. +-commutative 66.8%

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

      4. associate-+r+ 66.8%

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

      5. +-commutative 66.8%

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

      6. associate-+r+ 66.8%

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

      7. associate-+r+ 66.8%

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

      8. distribute-rgt1-in 66.8%

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

      9. +-commutative 66.8%

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

      10. *-commutative 66.8%

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

      11. distribute-rgt1-in 66.8%

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

      12. +-commutative 66.8%

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

      13. times-frac 91.6%

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

   3. Simplified 91.6%

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

      1. expm1-log1p-u 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 58.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 58.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-*r/ 91.7%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 86.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 7: 96.6% accurate, 2.7× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 10.0)
   (/ (/ 0.5 (+ alpha 3.0)) (+ alpha (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 10.0d0) then
        tmp = (0.5d0 / (alpha + 3.0d0)) / (alpha + (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 10.0) {
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 10.0:
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 10.0)
		tmp = Float64(Float64(0.5 / Float64(alpha + 3.0)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 10.0)
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 10.0], N[(N[(0.5 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 10

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 94.4%

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

      3. +-commutative 94.4%

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

      4. associate-+r+ 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.3%

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

      7. associate-+r+ 94.4%

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

      8. distribute-rgt1-in 94.4%

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

      9. +-commutative 94.4%

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

      10. *-commutative 94.4%

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

      11. distribute-rgt1-in 94.3%

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

      12. +-commutative 94.3%

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

      13. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 77.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r/ 98.6%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in alpha around 0 80.2%

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

   11. Taylor expanded in beta around 0 79.6%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{3 + \alpha}}}{\alpha + \left(2 + \beta\right)} \]
   12. Step-by-step derivation

       1. +-commutative 79.6%

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

   13. Simplified 79.6%

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

   if 10 < beta

   1. Initial program 86.6%

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

   2. Taylor expanded in beta around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

      3. sub-neg 81.7%

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

      4. mul-1-neg 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. +-commutative 81.7%

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

      7. mul-1-neg 81.7%

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

      8. distribute-lft-in 81.7%

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

      9. metadata-eval 81.7%

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

      10. mul-1-neg 81.7%

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

      11. unsub-neg 81.7%

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

   4. Simplified 81.7%

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

   5. Taylor expanded in beta around inf 81.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 3}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 8: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 11:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 11.0)
     (/ (/ 0.5 (+ alpha 3.0)) t_0)
     (/ (/ (+ 1.0 alpha) beta) t_0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 11.0) {
		tmp = (0.5 / (alpha + 3.0)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (2.0d0 + beta)
    if (beta <= 11.0d0) then
        tmp = (0.5d0 / (alpha + 3.0d0)) / t_0
    else
        tmp = ((1.0d0 + alpha) / beta) / t_0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 11.0) {
		tmp = (0.5 / (alpha + 3.0)) / t_0;
	} else {
		tmp = ((1.0 + alpha) / beta) / t_0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 11.0:
		tmp = (0.5 / (alpha + 3.0)) / t_0
	else:
		tmp = ((1.0 + alpha) / beta) / t_0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 11.0)
		tmp = Float64(Float64(0.5 / Float64(alpha + 3.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / t_0);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 11.0)
		tmp = (0.5 / (alpha + 3.0)) / t_0;
	else
		tmp = ((1.0 + alpha) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 11.0], N[(N[(0.5 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 11

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 94.4%

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

      3. +-commutative 94.4%

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

      4. associate-+r+ 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.3%

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

      7. associate-+r+ 94.4%

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

      8. distribute-rgt1-in 94.4%

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

      9. +-commutative 94.4%

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

      10. *-commutative 94.4%

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

      11. distribute-rgt1-in 94.3%

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

      12. +-commutative 94.3%

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

      13. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 77.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r/ 98.6%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in alpha around 0 80.2%

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

   11. Taylor expanded in beta around 0 79.6%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{3 + \alpha}}}{\alpha + \left(2 + \beta\right)} \]
   12. Step-by-step derivation

       1. +-commutative 79.6%

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

   13. Simplified 79.6%

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

   if 11 < beta

   1. Initial program 86.6%

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

      1. associate-/l/ 80.6%

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

      2. associate-/r* 70.5%

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

      3. +-commutative 70.5%

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

      4. associate-+r+ 70.5%

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

      5. +-commutative 70.5%

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

      6. associate-+r+ 70.5%

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

      7. associate-+r+ 70.5%

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

      8. distribute-rgt1-in 70.5%

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

      9. +-commutative 70.5%

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

      10. *-commutative 70.5%

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

      11. distribute-rgt1-in 70.5%

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

      12. +-commutative 70.5%

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

      13. times-frac 92.8%

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

   3. Simplified 92.8%

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

      1. expm1-log1p-u 92.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 52.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 52.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 52.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 92.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 92.8%

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

      3. *-commutative 92.8%

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

      4. associate-*r/ 92.8%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in beta around inf 81.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 3}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(2 + \beta\right)}\\ \end{array} \]

Alternative 9: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 3}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6)
   (/ (/ 0.5 (+ alpha 3.0)) (+ alpha (+ 2.0 beta)))
   (/ (/ (- alpha -1.0) beta) (+ beta (+ alpha 3.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 7.6d0) then
        tmp = (0.5d0 / (alpha + 3.0d0)) / (alpha + (2.0d0 + beta))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (beta + (alpha + 3.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6) {
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	} else {
		tmp = ((alpha - -1.0) / beta) / (beta + (alpha + 3.0));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 7.6:
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta))
	else:
		tmp = ((alpha - -1.0) / beta) / (beta + (alpha + 3.0))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6)
		tmp = Float64(Float64(0.5 / Float64(alpha + 3.0)) / Float64(alpha + Float64(2.0 + beta)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(beta + Float64(alpha + 3.0)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 7.6)
		tmp = (0.5 / (alpha + 3.0)) / (alpha + (2.0 + beta));
	else
		tmp = ((alpha - -1.0) / beta) / (beta + (alpha + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6], N[(N[(0.5 / N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5999999999999996

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 94.4%

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

      3. +-commutative 94.4%

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

      4. associate-+r+ 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.3%

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

      7. associate-+r+ 94.4%

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

      8. distribute-rgt1-in 94.4%

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

      9. +-commutative 94.4%

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

      10. *-commutative 94.4%

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

      11. distribute-rgt1-in 94.3%

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

      12. +-commutative 94.3%

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

      13. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)\right)} \]

      2. expm1-udef 77.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\right)} - 1} \]

      3. *-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}\right)} - 1 \]

      4. +-commutative 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}\right)} - 1 \]

   5. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)\right)} \]

      2. expm1-log1p 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r/ 98.6%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+r+ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in alpha around 0 80.2%

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

   11. Taylor expanded in beta around 0 79.6%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{3 + \alpha}}}{\alpha + \left(2 + \beta\right)} \]
   12. Step-by-step derivation

       1. +-commutative 79.6%

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

   13. Simplified 79.6%

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

   if 7.5999999999999996 < beta

   1. Initial program 86.6%

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

   2. Taylor expanded in beta around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

      3. sub-neg 81.7%

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

      4. mul-1-neg 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. +-commutative 81.7%

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

      7. mul-1-neg 81.7%

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

      8. distribute-lft-in 81.7%

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

      9. metadata-eval 81.7%

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

      10. mul-1-neg 81.7%

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

      11. unsub-neg 81.7%

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

   4. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. associate-/l/ 82.4%

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

      3. metadata-eval 82.4%

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

      4. associate-+l+ 82.4%

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

      5. metadata-eval 82.4%

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

      6. associate-+l+ 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. times-frac 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r/ 81.7%

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

      5. *-rgt-identity 81.7%

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

      6. +-commutative 81.7%

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

      7. +-commutative 81.7%

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

      8. +-commutative 81.7%

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

      9. associate-+r+ 81.7%

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

      10. +-commutative 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 3}}{\alpha + \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 10: 96.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 11.5)
   (/ 0.5 (* (+ alpha 3.0) (+ alpha 2.0)))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 11.5) {
		tmp = 0.5 / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 11.5d0) then
        tmp = 0.5d0 / ((alpha + 3.0d0) * (alpha + 2.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 11.5) {
		tmp = 0.5 / ((alpha + 3.0) * (alpha + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 11.5:
		tmp = 0.5 / ((alpha + 3.0) * (alpha + 2.0))
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 11.5)
		tmp = Float64(0.5 / Float64(Float64(alpha + 3.0) * Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 11.5)
		tmp = 0.5 / ((alpha + 3.0) * (alpha + 2.0));
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 11.5], N[(0.5 / N[(N[(alpha + 3.0), $MachinePrecision] * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 11.5

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

      8. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in alpha around 0 80.1%

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

   5. Taylor expanded in beta around 0 79.6%

      \[\leadsto \color{blue}{\frac{0.5}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if 11.5 < beta

   1. Initial program 86.6%

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

   2. Taylor expanded in beta around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

      3. sub-neg 81.7%

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

      4. mul-1-neg 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. +-commutative 81.7%

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

      7. mul-1-neg 81.7%

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

      8. distribute-lft-in 81.7%

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

      9. metadata-eval 81.7%

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

      10. mul-1-neg 81.7%

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

      11. unsub-neg 81.7%

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

   4. Simplified 81.7%

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

   5. Taylor expanded in beta around inf 81.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 11.5:\\ \;\;\;\;\frac{0.5}{\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11: 96.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 9

   1. Initial program 99.9%

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

      1. associate-/l/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

      8. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in alpha around 0 80.1%

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

   5. Taylor expanded in beta around 0 79.6%

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

   6. Taylor expanded in alpha around 0 59.2%

      \[\leadsto \frac{0.5}{\color{blue}{\left(2 + \beta\right) \cdot \left(3 + \beta\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 59.2%

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

   8. Simplified 59.2%

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

   if 9 < beta

   1. Initial program 86.6%

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

   2. Taylor expanded in beta around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

      3. sub-neg 81.7%

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

      4. mul-1-neg 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. +-commutative 81.7%

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

      7. mul-1-neg 81.7%

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

      8. distribute-lft-in 81.7%

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

      9. metadata-eval 81.7%

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

      10. mul-1-neg 81.7%

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

      11. unsub-neg 81.7%

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

   4. Simplified 81.7%

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

   5. Taylor expanded in beta around inf 81.4%

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

4. Final simplification 66.3%

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

Alternative 12: 52.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around -inf 35.5%

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

      1. associate-*r/ 35.5%

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

      2. mul-1-neg 35.5%

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

      3. sub-neg 35.5%

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

      4. mul-1-neg 35.5%

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

      5. distribute-neg-in 35.5%

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

      6. +-commutative 35.5%

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

      7. mul-1-neg 35.5%

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

      8. distribute-lft-in 35.5%

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

      9. metadata-eval 35.5%

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

      10. mul-1-neg 35.5%

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

      11. unsub-neg 35.5%

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

   4. Simplified 35.5%

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

   5. Taylor expanded in alpha around 0 34.4%

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

   if 1 < alpha

   1. Initial program 88.3%

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

      1. associate-/l/ 82.1%

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

      2. associate-/r* 72.3%

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

      3. +-commutative 72.3%

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

      4. associate-+r+ 72.3%

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

      5. +-commutative 72.3%

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

      6. associate-+r+ 72.3%

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

      7. associate-+r+ 72.3%

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

      8. distribute-rgt1-in 72.3%

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

      9. +-commutative 72.3%

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

      10. *-commutative 72.3%

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

      11. distribute-rgt1-in 72.3%

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

      12. +-commutative 72.3%

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

      13. times-frac 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. unpow2 13.6%

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

   6. Simplified 13.6%

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

   7. Taylor expanded in alpha around inf 13.6%

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

      1. unpow2 13.6%

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

   9. Simplified 13.6%

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

4. Final simplification 26.7%

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

Alternative 13: 53.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.4e-18)
   (/ (/ 1.0 beta) (+ beta 3.0))
   (/ (+ 1.0 alpha) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.4e-18) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.4d-18) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.4e-18) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if alpha <= 1.4e-18:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.4e-18)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.4e-18)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[alpha, 1.4e-18], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.40000000000000006e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in beta around -inf 35.5%

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

      1. associate-*r/ 35.5%

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

      2. mul-1-neg 35.5%

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

      3. sub-neg 35.5%

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

      4. mul-1-neg 35.5%

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

      5. distribute-neg-in 35.5%

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

      6. +-commutative 35.5%

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

      7. mul-1-neg 35.5%

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

      8. distribute-lft-in 35.5%

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

      9. metadata-eval 35.5%

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

      10. mul-1-neg 35.5%

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

      11. unsub-neg 35.5%

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

   4. Simplified 35.5%

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

   5. Taylor expanded in alpha around 0 34.6%

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

      1. associate-/r* 35.5%

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

      2. +-commutative 35.5%

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

   7. Simplified 35.5%

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

   if 1.40000000000000006e-18 < alpha

   1. Initial program 88.9%

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

      1. associate-/l/ 83.1%

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

      2. associate-/r* 73.9%

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

      3. +-commutative 73.9%

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

      4. associate-+r+ 73.9%

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

      5. +-commutative 73.9%

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

      6. associate-+r+ 73.9%

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

      7. associate-+r+ 73.9%

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

      8. distribute-rgt1-in 73.9%

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

      9. +-commutative 73.9%

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

      10. *-commutative 73.9%

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

      11. distribute-rgt1-in 73.9%

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

      12. +-commutative 73.9%

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

      13. times-frac 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in beta around inf 14.9%

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

      1. unpow2 14.9%

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

   6. Simplified 14.9%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14: 52.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 1

   1. Initial program 99.8%

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

      1. associate-/l/ 99.0%

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

      2. associate-/r* 95.1%

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

      3. +-commutative 95.1%

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

      4. associate-+r+ 95.1%

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

      5. +-commutative 95.1%

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

      6. associate-+r+ 95.1%

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

      7. associate-+r+ 95.1%

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

      8. distribute-rgt1-in 95.1%

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

      9. +-commutative 95.1%

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

      10. *-commutative 95.1%

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

      11. distribute-rgt1-in 95.1%

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

      12. +-commutative 95.1%

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

      13. times-frac 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in beta around inf 35.5%

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

      1. unpow2 35.5%

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

   6. Simplified 35.5%

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

   7. Taylor expanded in alpha around 0 35.2%

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

      1. unpow2 35.2%

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

   9. Simplified 35.2%

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

   if 1 < alpha

   1. Initial program 88.3%

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

      1. associate-/l/ 82.1%

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

      2. associate-/r* 72.3%

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

      3. +-commutative 72.3%

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

      4. associate-+r+ 72.3%

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

      5. +-commutative 72.3%

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

      6. associate-+r+ 72.3%

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

      7. associate-+r+ 72.3%

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

      8. distribute-rgt1-in 72.3%

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

      9. +-commutative 72.3%

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

      10. *-commutative 72.3%

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

      11. distribute-rgt1-in 72.3%

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

      12. +-commutative 72.3%

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

      13. times-frac 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in beta around inf 13.6%

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

      1. unpow2 13.6%

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

   6. Simplified 13.6%

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

   7. Taylor expanded in alpha around inf 13.6%

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

      1. unpow2 13.6%

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

   9. Simplified 13.6%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15: 52.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 86.7%

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

   3. +-commutative 86.7%

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

   4. associate-+r+ 86.7%

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

   5. +-commutative 86.7%

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

   6. associate-+r+ 86.7%

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

   7. associate-+r+ 86.7%

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

   8. distribute-rgt1-in 86.7%

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

   9. +-commutative 86.7%

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

   10. *-commutative 86.7%

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

   11. distribute-rgt1-in 86.7%

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

   12. +-commutative 86.7%

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

   13. times-frac 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in beta around inf 27.5%

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

   1. unpow2 27.5%

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

6. Simplified 27.5%

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

7. Final simplification 27.5%

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

Alternative 16: 55.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

2. Taylor expanded in beta around -inf 27.9%

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

   1. associate-*r/ 27.9%

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

   2. mul-1-neg 27.9%

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

   3. sub-neg 27.9%

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

   4. mul-1-neg 27.9%

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

   5. distribute-neg-in 27.9%

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

   6. +-commutative 27.9%

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

   7. mul-1-neg 27.9%

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

   8. distribute-lft-in 27.9%

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

   9. metadata-eval 27.9%

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

   10. mul-1-neg 27.9%

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

   11. unsub-neg 27.9%

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

4. Simplified 27.9%

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

5. Taylor expanded in beta around inf 28.3%

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

6. Final simplification 28.3%

   \[\leadsto \frac{\frac{\alpha - -1}{\beta}}{\beta} \]

Alternative 17: 33.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \frac{0.5}{\beta \cdot \beta} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 (/ 0.5 (* beta beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.5 / (beta * beta);
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0 / (beta * beta)
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.5 / (beta * beta);
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.5 / (beta * beta)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return Float64(0.5 / Float64(beta * beta))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.5 / (beta * beta);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(0.5 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 92.8%

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

   2. associate-+l+ 92.8%

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

   3. +-commutative 92.8%

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

   4. *-commutative 92.8%

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

   5. associate-+l+ 92.8%

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

   6. +-commutative 92.8%

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

   7. +-commutative 92.8%

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

   8. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in alpha around 0 82.2%

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

5. Taylor expanded in beta around 0 72.2%

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

6. Taylor expanded in beta around inf 18.4%

   \[\leadsto \color{blue}{\frac{0.5}{{\beta}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 18.4%

      \[\leadsto \frac{0.5}{\color{blue}{\beta \cdot \beta}} \]

8. Simplified 18.4%

   \[\leadsto \color{blue}{\frac{0.5}{\beta \cdot \beta}} \]

9. Final simplification 18.4%

   \[\leadsto \frac{0.5}{\beta \cdot \beta} \]

Alternative 18: 50.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 92.8%

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

   2. associate-/r* 86.7%

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

   3. +-commutative 86.7%

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

   4. associate-+r+ 86.7%

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

   5. +-commutative 86.7%

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

   6. associate-+r+ 86.7%

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

   7. associate-+r+ 86.7%

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

   8. distribute-rgt1-in 86.7%

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

   9. +-commutative 86.7%

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

   10. *-commutative 86.7%

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

   11. distribute-rgt1-in 86.7%

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

   12. +-commutative 86.7%

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

   13. times-frac 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in beta around inf 27.5%

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

   1. unpow2 27.5%

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

6. Simplified 27.5%

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

7. Taylor expanded in alpha around 0 27.1%

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

   1. unpow2 27.1%

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

9. Simplified 27.1%

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

10. Final simplification 27.1%

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

Alternative 19: 2.5% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. associate-/l/ 92.8%

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

   2. associate-+l+ 92.8%

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

   3. +-commutative 92.8%

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

   4. *-commutative 92.8%

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

   5. associate-+l+ 92.8%

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

   6. +-commutative 92.8%

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

   7. +-commutative 92.8%

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

   8. +-commutative 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in beta around -inf 48.5%

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

   1. mul-1-neg 48.5%

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

   2. sub-neg 48.5%

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

   3. mul-1-neg 48.5%

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

   4. distribute-neg-in 48.5%

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

   5. +-commutative 48.5%

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

   6. mul-1-neg 48.5%

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

   7. distribute-lft-in 48.5%

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

   8. metadata-eval 48.5%

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

   9. mul-1-neg 48.5%

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

   10. unsub-neg 48.5%

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

6. Simplified 48.5%

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

7. Taylor expanded in alpha around inf 4.5%

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

8. Final simplification 4.5%

   \[\leadsto \frac{1}{\alpha} \]

Alternative 20: 6.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

2. Taylor expanded in beta around -inf 27.9%

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

   1. associate-*r/ 27.9%

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

   2. mul-1-neg 27.9%

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

   3. sub-neg 27.9%

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

   4. mul-1-neg 27.9%

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

   5. distribute-neg-in 27.9%

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

   6. +-commutative 27.9%

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

   7. mul-1-neg 27.9%

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

   8. distribute-lft-in 27.9%

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

   9. metadata-eval 27.9%

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

   10. mul-1-neg 27.9%

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

   11. unsub-neg 27.9%

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

4. Simplified 27.9%

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

5. Taylor expanded in alpha around inf 3.9%

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

6. Final simplification 3.9%

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

Reproduce

herbie shell --seed 2023275 
(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)))