Octave 3.8, jcobi/1

Average Error: 16.1 → 0.1

Time: 10.8s

Precision: binary64

Cost: 8388

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
   (/
    (fma
     (/ (- (- -2.0 beta) beta) alpha)
     (/ (+ beta 2.0) alpha)
     (/ (+ 2.0 (+ beta beta)) alpha))
    2.0)
   (/ (fma (/ 1.0 (+ beta (+ alpha 2.0))) (- beta alpha) 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((2.0 + (beta + beta)) / alpha)) / 2.0;
	} else {
		tmp = fma((1.0 / (beta + (alpha + 2.0))), (beta - alpha), 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		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(fma(Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), Float64(beta - alpha), 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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], 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[(N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.998999999999999999

   1. Initial program 59.2

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

   2. Taylor expanded in alpha around -inf 3.3

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

   3. Simplified 0.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{2 + \beta}{\alpha}, \frac{\left(\beta + \beta\right) + 2}{\alpha}\right)}}{2} \]
      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)): 3 points increase in error, 4 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))): 3 points increase in error, 1 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)): 34 points increase in error, 20 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 -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{t_1 + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	19.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 0.0021:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	16.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1350:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot -0.5\\ \end{array} \]

Alternative 7
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 490:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	19.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	19.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 1700000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	20.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 31.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 11
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022321 
(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))