Octave 3.8, jcobi/3

Percentage Accurate: 94.9% → 99.8%

Time: 25.7s

Alternatives: 18

Speedup: 1.4×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 94.9% 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, 1.3× speedup

Math

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

FPCore

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

C

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

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)
    code = (((1.0d0 + alpha) * ((1.0d0 + beta) / t_0)) / t_0) / (1.0d0 + (2.0d0 + (alpha + beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

   1. div-inv 95.2%

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

   2. +-commutative 95.2%

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

   3. associate-+l+ 95.2%

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

   4. *-commutative 95.2%

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

   5. metadata-eval 95.2%

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

   6. associate-+r+ 95.2%

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

   7. metadata-eval 95.2%

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

   8. associate-+r+ 95.2%

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

3. Applied egg-rr 95.2%

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

   1. associate-*l/ 95.2%

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

   2. associate-*r/ 95.2%

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

   3. *-rgt-identity 95.2%

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

   4. associate-+r+ 95.2%

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

   5. *-rgt-identity 95.2%

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

   6. +-commutative 95.2%

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

   7. distribute-rgt1-in 95.2%

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

   8. distribute-lft-in 95.2%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\beta \cdot \left(\beta + 3\right) + 2 \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.6e-14)
   (*
    (+ 1.0 beta)
    (/
     (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
     (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (+ 1.0 alpha)
    (/
     (/ (+ 1.0 beta) (+ beta 2.0))
     (+ (* beta (+ beta 3.0)) (* 2.0 (+ beta 3.0)))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta * (beta + 3.0)) + (2.0 * (beta + 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.6d-14) then
        tmp = (1.0d0 + beta) * (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / (beta + 2.0d0)) / ((beta * (beta + 3.0d0)) + (2.0d0 * (beta + 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta * (beta + 3.0)) + (2.0 * (beta + 3.0))));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.6e-14:
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta * (beta + 3.0)) + (2.0 * (beta + 3.0))))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.6e-14)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta * Float64(beta + 3.0)) + Float64(2.0 * Float64(beta + 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.6e-14)
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta * (beta + 3.0)) + (2.0 * (beta + 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.6e-14], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.59999999999999996e-14

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*r/ 99.7%

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

      11. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 8.59999999999999996e-14 < beta

   1. Initial program 84.5%

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

      1. associate-/l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. +-commutative 82.9%

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

      4. associate-+r+ 82.9%

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

      5. associate-+l+ 82.9%

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

      6. distribute-rgt1-in 82.9%

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

      7. *-rgt-identity 82.9%

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

      8. distribute-lft-out 82.9%

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

      9. +-commutative 82.9%

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

      10. associate-*l/ 96.1%

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

      11. *-commutative 96.1%

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

      12. associate-*r/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in alpha around 0 84.8%

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

      1. distribute-lft-in 84.8%

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

   6. Applied egg-rr 84.8%

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

   7. Taylor expanded in alpha around 0 82.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\beta \cdot \left(\beta + 3\right) + 2 \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3: 93.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) (* t_0 (+ alpha (+ beta 3.0)))))))

C

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

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)
    code = (1.0d0 + alpha) * (((1.0d0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0))));
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

   1. associate-/l/ 94.7%

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

   2. associate-+l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. associate-+r+ 94.7%

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

   5. associate-+l+ 94.7%

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

   6. distribute-rgt1-in 94.7%

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

   7. *-rgt-identity 94.7%

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

   8. distribute-lft-out 94.7%

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

   9. +-commutative 94.7%

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

   10. associate-*l/ 98.6%

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

   11. *-commutative 98.6%

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

   12. associate-*r/ 96.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (/ (/ (+ 1.0 beta) t_0) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((1.0 + alpha) / (alpha + (beta + 3.0))) * (((1.0 + beta) / t_0) / t_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)
    code = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) * (((1.0d0 + beta) / t_0) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (((1.0 + beta) / t_0) / t_0);
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

   1. div-inv 95.2%

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

   2. +-commutative 95.2%

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

   3. associate-+l+ 95.2%

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

   4. *-commutative 95.2%

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

   5. metadata-eval 95.2%

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

   6. associate-+r+ 95.2%

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

   7. metadata-eval 95.2%

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

   8. associate-+r+ 95.2%

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

3. Applied egg-rr 95.2%

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

   1. associate-*l/ 95.2%

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

   2. associate-*r/ 95.2%

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

   3. *-rgt-identity 95.2%

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

   4. associate-+r+ 95.2%

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

   5. *-rgt-identity 95.2%

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

   6. +-commutative 95.2%

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

   7. distribute-rgt1-in 95.2%

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

   8. distribute-lft-in 95.2%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 76.8%

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

7. Applied egg-rr 76.8%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

   3. times-frac 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ 1.0 beta) t_0) (/ (/ (+ 1.0 alpha) t_0) (+ (+ alpha beta) 3.0)))))

C

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

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)
    code = ((1.0d0 + beta) / t_0) * (((1.0d0 + alpha) / t_0) / ((alpha + beta) + 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((1.0 + beta) / t_0) * (((1.0 + alpha) / t_0) / ((alpha + beta) + 3.0));
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

   1. associate-/l/ 94.7%

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

   2. associate-+l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. associate-+r+ 94.7%

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

   5. associate-+l+ 94.7%

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

   6. distribute-rgt1-in 94.7%

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

   7. *-rgt-identity 94.7%

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

   8. distribute-lft-out 94.7%

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

   9. +-commutative 94.7%

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

   10. associate-*l/ 98.6%

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

   11. *-commutative 98.6%

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

   12. associate-*r/ 96.1%

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

3. Simplified 96.1%

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

   1. associate-*r/ 98.6%

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

   2. +-commutative 98.6%

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

5. Applied egg-rr 98.6%

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

   1. +-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. +-commutative 98.6%

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

   4. associate-*r/ 98.6%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-+r+ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2e-15)
   (*
    (+ 1.0 beta)
    (/
     (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
     (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (+ 1.0 alpha)
    (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e-15) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d-15) then
        tmp = (1.0d0 + beta) * (((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e-15) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.2e-15:
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2e-15)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2e-15)
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + (beta + 2.0))) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.2e-15], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999997e-15

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*r/ 99.7%

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

      11. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 1.19999999999999997e-15 < beta

   1. Initial program 84.6%

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   2. 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-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. associate-+r+ 83.1%

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

      5. associate-+l+ 83.1%

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

      6. distribute-rgt1-in 83.1%

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

      7. *-rgt-identity 83.1%

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

      8. distribute-lft-out 83.1%

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

      9. +-commutative 83.1%

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

      10. associate-*l/ 96.1%

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

      11. *-commutative 96.1%

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

      12. associate-*r/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in alpha around 0 83.8%

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

   5. Taylor expanded in alpha around 0 81.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{t_0}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\frac{t_0}{1 + \beta}}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 9e-14)
     (* (+ 1.0 beta) (/ (/ (+ 1.0 alpha) t_0) (* (+ alpha 2.0) (+ alpha 3.0))))
     (/
      (/ (+ 1.0 alpha) (/ t_0 (+ 1.0 beta)))
      (* (+ beta 2.0) (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / t_0) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (t_0 / (1.0 + beta))) / ((beta + 2.0) * (beta + 3.0));
	}
	return tmp;
}

Fortran

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 + (beta + 2.0d0)
    if (beta <= 9d-14) then
        tmp = (1.0d0 + beta) * (((1.0d0 + alpha) / t_0) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (t_0 / (1.0d0 + beta))) / ((beta + 2.0d0) * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 9e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / t_0) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (t_0 / (1.0 + beta))) / ((beta + 2.0) * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 9e-14:
		tmp = (1.0 + beta) * (((1.0 + alpha) / t_0) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (t_0 / (1.0 + beta))) / ((beta + 2.0) * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 9e-14)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(t_0 / Float64(1.0 + beta))) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 9e-14)
		tmp = (1.0 + beta) * (((1.0 + alpha) / t_0) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = ((1.0 + alpha) / (t_0 / (1.0 + beta))) / ((beta + 2.0) * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e-14], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.9999999999999995e-14

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*r/ 99.7%

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

      11. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 8.9999999999999995e-14 < beta

   1. Initial program 84.5%

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

      1. associate-/l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. +-commutative 82.9%

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

      4. associate-+r+ 82.9%

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

      5. associate-+l+ 82.9%

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

      6. distribute-rgt1-in 82.9%

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

      7. *-rgt-identity 82.9%

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

      8. distribute-lft-out 82.9%

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

      9. +-commutative 82.9%

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

      10. associate-*l/ 96.1%

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

      11. *-commutative 96.1%

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

      12. associate-*r/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in alpha around 0 84.8%

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

      1. expm1-log1p-u 84.8%

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

      2. expm1-udef 59.6%

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

      3. +-commutative 59.6%

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

      4. +-commutative 59.6%

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

      5. *-commutative 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. expm1-def 84.8%

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

      2. expm1-log1p 84.8%

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

      3. associate-*r/ 82.3%

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

      4. associate-*r/ 71.7%

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

      5. associate-/l* 82.3%

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

      6. +-commutative 82.3%

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

      7. +-commutative 82.3%

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

      8. *-commutative 82.3%

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

   8. Simplified 82.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\frac{\alpha + \left(\beta + 2\right)}{1 + \beta}}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.5e-14)
   (*
    (+ 1.0 beta)
    (/ (/ (+ 1.0 alpha) (+ alpha 2.0)) (* (+ alpha 2.0) (+ alpha 3.0))))
   (*
    (+ 1.0 alpha)
    (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.5d-14) then
        tmp = (1.0d0 + beta) * (((1.0d0 + alpha) / (alpha + 2.0d0)) / ((alpha + 2.0d0) * (alpha + 3.0d0)))
    else
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.5e-14) {
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0)));
	} else {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.5e-14:
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0)))
	else:
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.5e-14)
		tmp = Float64(Float64(1.0 + beta) * Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) / Float64(Float64(alpha + 2.0) * Float64(alpha + 3.0))));
	else
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.5e-14)
		tmp = (1.0 + beta) * (((1.0 + alpha) / (alpha + 2.0)) / ((alpha + 2.0) * (alpha + 3.0)));
	else
		tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.5e-14], N[(N[(1.0 + beta), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] * N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000001e-14

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*r/ 99.7%

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

      11. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in beta around 0 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in beta around 0 98.5%

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

   if 6.5000000000000001e-14 < beta

   1. Initial program 84.5%

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

      1. associate-/l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. +-commutative 82.9%

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

      4. associate-+r+ 82.9%

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

      5. associate-+l+ 82.9%

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

      6. distribute-rgt1-in 82.9%

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

      7. *-rgt-identity 82.9%

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

      8. distribute-lft-out 82.9%

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

      9. +-commutative 82.9%

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

      10. associate-*l/ 96.1%

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

      11. *-commutative 96.1%

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

      12. associate-*r/ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in alpha around 0 84.8%

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

   5. Taylor expanded in alpha around 0 82.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6e+15)
   (/ (+ 1.0 beta) (* (+ alpha (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+15) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d+15) then
        tmp = (1.0d0 + beta) / ((alpha + (beta + 2.0d0)) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6e+15) {
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6e+15:
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(alpha + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6e+15)
		tmp = (1.0 + beta) / ((alpha + (beta + 2.0)) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.6e15

   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/ 99.7%

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

      2. associate-/r* 95.4%

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

      3. associate-+l+ 95.4%

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

      4. +-commutative 95.4%

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

      5. associate-+r+ 95.4%

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

      6. associate-+l+ 95.4%

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

      7. distribute-rgt1-in 95.4%

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

      8. *-rgt-identity 95.4%

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

      9. distribute-lft-out 95.4%

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

      10. *-commutative 95.4%

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

      11. metadata-eval 95.4%

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

      12. associate-+l+ 95.4%

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

      13. +-commutative 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in alpha around 0 84.0%

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

   5. Taylor expanded in alpha around 0 68.6%

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

   if 3.6e15 < beta

   1. Initial program 81.4%

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

      1. div-inv 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-+l+ 81.4%

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

      4. *-commutative 81.4%

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

      5. metadata-eval 81.4%

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

      6. associate-+r+ 81.4%

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

      7. metadata-eval 81.4%

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

      8. associate-+r+ 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. associate-*l/ 81.4%

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

      2. associate-*r/ 81.4%

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

      3. *-rgt-identity 81.4%

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

      4. associate-+r+ 81.4%

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

      5. *-rgt-identity 81.4%

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

      6. +-commutative 81.4%

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

      7. distribute-rgt1-in 81.4%

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

      8. distribute-lft-in 81.4%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in beta around inf 80.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 10: 71.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (*
  (+ 1.0 alpha)
  (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* (+ beta 2.0) (+ beta 3.0)))))

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + alpha) * (((1.0d0 + beta) / (beta + 2.0d0)) / ((beta + 2.0d0) * (beta + 3.0d0)))
end function

Java

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

Python

def code(alpha, beta):
	return (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)))

Julia

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

MATLAB

function tmp = code(alpha, beta)
	tmp = (1.0 + alpha) * (((1.0 + beta) / (beta + 2.0)) / ((beta + 2.0) * (beta + 3.0)));
end

Wolfram

code[alpha_, beta_] := N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

   1. associate-/l/ 94.7%

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

   2. associate-+l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. associate-+r+ 94.7%

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

   5. associate-+l+ 94.7%

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

   6. distribute-rgt1-in 94.7%

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

   7. *-rgt-identity 94.7%

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

   8. distribute-lft-out 94.7%

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

   9. +-commutative 94.7%

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

   10. associate-*l/ 98.6%

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

   11. *-commutative 98.6%

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

   12. associate-*r/ 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in alpha around 0 72.7%

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

5. Taylor expanded in alpha around 0 71.6%

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

6. Final simplification 71.6%

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

Alternative 11: 72.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.2e+50)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (* (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+50) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.2d+50) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.2e+50) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.2e+50:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.2e+50)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.2e+50)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.2e+50], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.20000000000000017e50

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in alpha around 0 66.8%

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

      1. distribute-lft-in 66.8%

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

   6. Applied egg-rr 66.8%

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

   7. Taylor expanded in alpha around 0 66.4%

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

   8. Taylor expanded in alpha around 0 66.9%

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

      1. distribute-rgt-in 66.9%

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

      2. *-commutative 66.9%

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

   10. Simplified 66.9%

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

   if 2.20000000000000017e50 < beta

   1. Initial program 79.2%

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

      1. div-inv 79.1%

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

      2. +-commutative 79.1%

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

      3. associate-+l+ 79.1%

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

      4. *-commutative 79.1%

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

      5. metadata-eval 79.1%

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

      6. associate-+r+ 79.1%

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

      7. metadata-eval 79.1%

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

      8. associate-+r+ 79.1%

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

   3. Applied egg-rr 79.1%

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

      1. associate-*l/ 79.2%

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

      2. associate-*r/ 79.2%

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

      3. *-rgt-identity 79.2%

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

      4. associate-+r+ 79.2%

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

      5. *-rgt-identity 79.2%

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

      6. +-commutative 79.2%

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

      7. distribute-rgt1-in 79.2%

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

      8. distribute-lft-in 79.2%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 60.1%

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

   7. Applied egg-rr 60.1%

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

      1. expm1-def 94.8%

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

      2. expm1-log1p 94.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in beta around inf 85.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12: 72.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+15)
   (/ (+ 1.0 beta) (* (+ beta 2.0) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d+15) then
        tmp = (1.0d0 + beta) / ((beta + 2.0d0) * ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+15) {
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e+15:
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+15)
		tmp = Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e+15)
		tmp = (1.0 + beta) / ((beta + 2.0) * ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5e+15], N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5e15

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in alpha around 0 67.6%

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

      1. distribute-lft-in 67.6%

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

   6. Applied egg-rr 67.6%

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

   7. Taylor expanded in alpha around 0 67.2%

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

   8. Taylor expanded in alpha around 0 67.7%

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

      1. distribute-rgt-in 67.7%

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

      2. *-commutative 67.7%

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

   10. Simplified 67.7%

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

   if 1.5e15 < beta

   1. Initial program 81.4%

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

      1. div-inv 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-+l+ 81.4%

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

      4. *-commutative 81.4%

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

      5. metadata-eval 81.4%

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

      6. associate-+r+ 81.4%

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

      7. metadata-eval 81.4%

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

      8. associate-+r+ 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. associate-*l/ 81.4%

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

      2. associate-*r/ 81.4%

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

      3. *-rgt-identity 81.4%

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

      4. associate-+r+ 81.4%

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

      5. *-rgt-identity 81.4%

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

      6. +-commutative 81.4%

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

      7. distribute-rgt1-in 81.4%

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

      8. distribute-lft-in 81.4%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in beta around inf 80.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 13: 72.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.8e+47)
   (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) (+ beta 2.0))
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+47) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.8d+47) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / (beta + 2.0d0)
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (alpha + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.8e+47) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.8e+47:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0)
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.8e+47)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / Float64(beta + 2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.8e+47)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + 2.0);
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.8e+47], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.80000000000000004e47

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in alpha around 0 66.8%

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

      1. distribute-lft-in 66.8%

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

   6. Applied egg-rr 66.8%

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

   7. Taylor expanded in alpha around 0 66.9%

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

      1. associate-/r* 66.9%

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

      2. distribute-rgt-in 66.9%

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

   9. Simplified 66.9%

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

   if 1.80000000000000004e47 < beta

   1. Initial program 79.2%

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

      1. div-inv 79.1%

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

      2. +-commutative 79.1%

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

      3. associate-+l+ 79.1%

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

      4. *-commutative 79.1%

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

      5. metadata-eval 79.1%

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

      6. associate-+r+ 79.1%

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

      7. metadata-eval 79.1%

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

      8. associate-+r+ 79.1%

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

   3. Applied egg-rr 79.1%

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

      1. associate-*l/ 79.2%

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

      2. associate-*r/ 79.2%

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

      3. *-rgt-identity 79.2%

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

      4. associate-+r+ 79.2%

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

      5. *-rgt-identity 79.2%

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

      6. +-commutative 79.2%

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

      7. distribute-rgt1-in 79.2%

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

      8. distribute-lft-in 79.2%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in beta around inf 85.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 14: 71.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (* (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0))) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.5d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. distribute-lft-in 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in alpha around 0 67.4%

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

   8. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   10. Simplified 66.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.5 < beta

   1. Initial program 82.7%

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

      1. div-inv 82.7%

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

      2. +-commutative 82.7%

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

      3. associate-+l+ 82.7%

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

      4. *-commutative 82.7%

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

      5. metadata-eval 82.7%

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

      6. associate-+r+ 82.7%

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

      7. metadata-eval 82.7%

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

      8. associate-+r+ 82.7%

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

   3. Applied egg-rr 82.7%

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

      1. associate-*l/ 82.7%

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

      2. associate-*r/ 82.7%

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

      3. *-rgt-identity 82.7%

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

      4. associate-+r+ 82.7%

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

      5. *-rgt-identity 82.7%

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

      6. +-commutative 82.7%

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

      7. distribute-rgt1-in 82.7%

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

      8. distribute-lft-in 82.7%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. expm1-def 95.7%

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

      2. expm1-log1p 95.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in beta around inf 77.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 15: 70.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ (+ 1.0 alpha) (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = (1.0d0 + alpha) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = (1.0 + alpha) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = (1.0 + alpha) / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = (1.0 + alpha) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. distribute-lft-in 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in alpha around 0 67.4%

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

   8. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   10. Simplified 66.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 82.7%

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

      1. div-inv 82.7%

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

      2. +-commutative 82.7%

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

      3. associate-+l+ 82.7%

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

      4. *-commutative 82.7%

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

      5. metadata-eval 82.7%

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

      6. associate-+r+ 82.7%

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

      7. metadata-eval 82.7%

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

      8. associate-+r+ 82.7%

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

   3. Applied egg-rr 82.7%

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

      1. associate-*l/ 82.7%

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

      2. associate-*r/ 82.7%

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

      3. *-rgt-identity 82.7%

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

      4. associate-+r+ 82.7%

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

      5. *-rgt-identity 82.7%

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

      6. +-commutative 82.7%

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

      7. distribute-rgt1-in 82.7%

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

      8. distribute-lft-in 82.7%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in beta around inf 80.9%

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

      1. unpow2 80.9%

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

   8. Simplified 80.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16: 68.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.5) 0.08333333333333333 (/ 1.0 (* alpha alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.5) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.5d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (alpha * alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.5) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.5:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (alpha * alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.5)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(alpha * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.5)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (alpha * alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3.5], 0.08333333333333333, N[(1.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.5

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*l/ 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in alpha around 0 97.8%

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

      1. distribute-lft-in 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in alpha around 0 94.7%

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

   8. Taylor expanded in beta around 0 71.2%

      \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3.5 < alpha

   1. Initial program 85.9%

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

      1. associate-/l/ 84.4%

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

      2. associate-/l/ 71.2%

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

      3. associate-+l+ 71.2%

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

      4. +-commutative 71.2%

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

      5. associate-+r+ 71.2%

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

      6. associate-+l+ 71.2%

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

      7. distribute-rgt1-in 71.2%

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

      8. *-rgt-identity 71.2%

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

      9. distribute-lft-out 71.2%

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

      10. +-commutative 71.2%

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

      11. times-frac 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in alpha around inf 85.5%

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

      1. +-commutative 85.5%

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

      2. unpow2 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in beta around 0 84.8%

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

      1. unpow2 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 17: 69.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (+ 0.08333333333333333 (* beta -0.027777777777777776))
   (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.08333333333333333d0 + (beta * (-0.027777777777777776d0))
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.08333333333333333 + Float64(beta * -0.027777777777777776));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.08333333333333333 + (beta * -0.027777777777777776);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.08333333333333333 + N[(beta * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   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/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. associate-+l+ 99.7%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*l/ 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*r/ 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in alpha around 0 67.3%

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

      1. distribute-lft-in 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in alpha around 0 67.4%

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

   8. Taylor expanded in beta around 0 66.3%

      \[\leadsto \color{blue}{0.08333333333333333 + -0.027777777777777776 \cdot \beta} \]
   9. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot -0.027777777777777776} \]

   10. Simplified 66.3%

       \[\leadsto \color{blue}{0.08333333333333333 + \beta \cdot -0.027777777777777776} \]

   if 2.7999999999999998 < beta

   1. Initial program 82.7%

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

      1. associate-/l/ 81.0%

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

      2. associate-+l+ 81.0%

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

      3. +-commutative 81.0%

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

      4. associate-+r+ 81.0%

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

      5. associate-+l+ 81.0%

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

      6. distribute-rgt1-in 81.0%

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

      7. *-rgt-identity 81.0%

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

      8. distribute-lft-out 81.0%

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

      9. +-commutative 81.0%

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

      10. associate-*l/ 95.7%

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

      11. *-commutative 95.7%

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

      12. associate-*r/ 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in alpha around 0 87.3%

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

      1. distribute-lft-in 87.3%

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in alpha around 0 71.2%

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

   8. Taylor expanded in beta around inf 74.3%

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

      1. unpow2 74.3%

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

   10. Simplified 74.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 18: 44.9% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.2%

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

   1. associate-/l/ 94.7%

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

   2. associate-+l+ 94.7%

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

   3. +-commutative 94.7%

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

   4. associate-+r+ 94.7%

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

   5. associate-+l+ 94.7%

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

   6. distribute-rgt1-in 94.7%

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

   7. *-rgt-identity 94.7%

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

   8. distribute-lft-out 94.7%

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

   9. +-commutative 94.7%

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

   10. associate-*l/ 98.6%

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

   11. *-commutative 98.6%

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

   12. associate-*r/ 96.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in alpha around 0 72.7%

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

   1. distribute-lft-in 72.7%

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

6. Applied egg-rr 72.7%

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

7. Taylor expanded in alpha around 0 68.4%

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

8. Taylor expanded in beta around 0 48.9%

   \[\leadsto \color{blue}{0.08333333333333333} \]

9. Final simplification 48.9%

   \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023187 
(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)))