Octave 3.8, jcobi/3

Average Error: 3.5 → 0.1

Time: 16.9s

Precision: binary64

Cost: 1600

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ 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)))

↓

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

C

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

↓

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

Fortran

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

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (2.0d0 + beta)
    code = ((1.0d0 + alpha) / (t_0 / ((1.0d0 + beta) / t_0))) / (beta + (alpha + 3.0d0))
end function

Java

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

↓

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

Python

def code(alpha, beta):
	return (((((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)

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 3.5

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

2. Simplified 9.9

   \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   Proof
   (/.f64 (*.f64 (+.f64 beta 1) (+.f64 alpha 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (*.f64 (+.f64 beta 1) 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (Rewrite=> *-rgt-identity_binary64 (+.f64 beta 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 beta alpha) alpha)) (+.f64 beta 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 beta alpha) alpha) beta) 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha beta))) 1) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha))) 1) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (+.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))))): 1 points increase in error, 1 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 3 points increase in error, 31 points decrease in error
   (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))): 14 points increase in error, 8 points decrease in error

3. Applied egg-rr 1.6

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Applied egg-rr 0.1

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

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 10^{+151}:\\ \;\;\;\;\frac{1 + \beta}{t_0 \cdot t_1} \cdot \frac{1 + \alpha}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)}}{t_1}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 95000000:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha \cdot 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\beta}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(6 + \beta \cdot \left(\beta + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{2 + \left(\alpha + \beta\right)}}{-3 - \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 8
Error	1.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot 0.0023148148148148147\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta} + \frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	1.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot 0.0023148148148148147\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 10
Error	2.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	1.9
Cost	708

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

Alternative 12
Error	1.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;0.08333333333333333 + \beta \cdot \left(-0.027777777777777776 + \beta \cdot 0.0023148148148148147\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Error	3.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	33.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]

Alternative 15
Error	5.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Error	5.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	34.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2}{\beta}\\ \end{array} \]

Alternative 18
Error	56.5
Cost	64

\[0.0625 \]

Alternative 19
Error	35.2
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2022329 
(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)))