Octave 3.8, jcobi/1

Average Error: 16.6 → 0.1

Time: 8.6s

Precision: binary64

Cost: 8388

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ 2.0 (+ beta beta)) alpha))
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

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

↓

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((2.0 + (beta + beta)) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9999)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(2.0 + Float64(beta + beta)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99990000000000001

   1. Initial program 59.4

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

   2. Taylor expanded in alpha around -inf 3.9

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

   3. Simplified 0.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{2 + \beta}{\alpha}, \frac{\left(\beta + \beta\right) + 2}{\alpha}\right)}}{2} \]
      Proof
      (fma.f64 (/.f64 (-.f64 (-.f64 -2 beta) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 2)) beta) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 2) (neg.f64 beta))) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 2 beta))) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta 2))) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2))) beta) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (+.f64 beta 2)) (neg.f64 beta))) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (+.f64 (*.f64 -1 (+.f64 beta 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta))) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta 2)))) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (+.f64 (*.f64 -1 beta) (Rewrite=> mul-1-neg_binary64 (neg.f64 (+.f64 beta 2)))) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2))) alpha) (/.f64 (+.f64 2 beta) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta 2)) alpha) (/.f64 (+.f64 (+.f64 beta beta) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 (+.f64 beta (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 beta)))) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 (+.f64 beta (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)))) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 beta (*.f64 -1 beta))) 2) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 (-.f64 beta (*.f64 -1 beta)) (Rewrite<= metadata-eval (neg.f64 -2))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 (-.f64 beta (*.f64 -1 beta)) (neg.f64 (Rewrite<= metadata-eval (neg.f64 2)))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 beta (*.f64 -1 beta)) (neg.f64 2))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 beta (+.f64 (*.f64 -1 beta) (neg.f64 2)))) alpha)): 1 points increase in error, 1 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (-.f64 beta (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 beta)) (neg.f64 2))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (-.f64 beta (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 beta 2)))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (-.f64 beta (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2)))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2)))))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (neg.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 beta (*.f64 (neg.f64 -1) (+.f64 beta 2)))))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (neg.f64 (+.f64 beta (*.f64 (Rewrite=> metadata-eval 1) (+.f64 beta 2))))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (neg.f64 (+.f64 beta (Rewrite=> *-lft-identity_binary64 (+.f64 beta 2))))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 beta) (neg.f64 (+.f64 beta 2))))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)) (neg.f64 (+.f64 beta 2)))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) alpha)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (/.f64 (+.f64 beta 2) alpha)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha))): 2 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) (+.f64 beta 2)) (*.f64 alpha alpha))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 31 points increase in error, 21 points decrease in error
      (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) (+.f64 beta 2)) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 -1 beta) (neg.f64 (+.f64 beta 2)))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (+.f64 (*.f64 -1 beta) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2)))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 -1 (+.f64 beta (+.f64 beta 2)))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (*.f64 -1 (+.f64 beta (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 beta 2))))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (*.f64 -1 (+.f64 beta (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 beta 2)))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (*.f64 -1 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))))) (+.f64 beta 2)) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (+.f64 beta 2)))) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (+.f64 beta 2)) (pow.f64 alpha 2)))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 beta 2) (-.f64 beta (*.f64 -1 (+.f64 beta 2))))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta 2) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 beta (*.f64 (neg.f64 -1) (+.f64 beta 2))))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta (*.f64 (Rewrite=> metadata-eval 1) (+.f64 beta 2)))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta (Rewrite=> *-lft-identity_binary64 (+.f64 beta 2)))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 beta (+.f64 beta 2)) (*.f64 (+.f64 beta 2) (+.f64 beta 2)))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (+.f64 (*.f64 beta (+.f64 beta 2)) (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2)))) (pow.f64 alpha 2))) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2))) (pow.f64 alpha 2))) (neg.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2))) (pow.f64 alpha 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 -1 (/.f64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2))) (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	7.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 450000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 820000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	18.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	19.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.75:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	18.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.68:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 7
Error	18.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 245:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	40.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022328 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))