Octave 3.8, jcobi/1

Average Error: 16.3 → 0.1

Time: 7.6s

Precision: binary64

Cost: 8516

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha} + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

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)) -0.9995)
   (/
    (fma
     2.0
     (/ beta alpha)
     (+
      (/ 2.0 alpha)
      (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) (* alpha 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)) <= -0.9995) {
		tmp = fma(2.0, (beta / alpha), ((2.0 / alpha) + ((beta + 2.0) * (((-2.0 - beta) - beta) / (alpha * 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)) <= -0.9995)
		tmp = Float64(fma(2.0, Float64(beta / alpha), Float64(Float64(2.0 / alpha) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / Float64(alpha * 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], -0.9995], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision] + N[(N[(2.0 / alpha), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.99950000000000006

   1. Initial program 59.3

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

   2. Taylor expanded in alpha around inf 3.0

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

   3. Simplified 0.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha} + \frac{\beta + \left(2 + \beta\right)}{\alpha \cdot \alpha} \cdot \left(-2 - \beta\right)\right)}}{2} \]
      Proof
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (/.f64 2 alpha) (*.f64 (/.f64 (+.f64 beta (+.f64 2 beta)) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) alpha) (*.f64 (/.f64 (+.f64 beta (+.f64 2 beta)) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 alpha))) (*.f64 (/.f64 (+.f64 beta (+.f64 2 beta)) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 beta))) (+.f64 2 beta)) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta))) (+.f64 2 beta)) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (+.f64 (neg.f64 (*.f64 -1 beta)) (Rewrite<= +-commutative_binary64 (+.f64 beta 2))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (+.f64 (neg.f64 (*.f64 -1 beta)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 beta 2))))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (+.f64 (neg.f64 (*.f64 -1 beta)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2))))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta 2))))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (neg.f64 (+.f64 (*.f64 -1 beta) (Rewrite=> mul-1-neg_binary64 (neg.f64 (+.f64 beta 2))))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) (*.f64 alpha alpha)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (*.f64 -1 (-.f64 (*.f64 -1 beta) (+.f64 beta 2))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 -1 beta) (neg.f64 (+.f64 beta 2))))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (neg.f64 (+.f64 (*.f64 -1 beta) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2))))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 -1 beta)) (neg.f64 (*.f64 -1 (+.f64 beta 2))))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (neg.f64 (*.f64 -1 beta)) (*.f64 -1 (+.f64 beta 2)))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 beta))) (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 (Rewrite=> remove-double-neg_binary64 beta) (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (-.f64 -2 beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (-.f64 (Rewrite<= metadata-eval (*.f64 2 -1)) beta)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 -1) (neg.f64 beta)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (+.f64 (Rewrite=> metadata-eval -2) (neg.f64 beta))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (+.f64 (Rewrite<= metadata-eval (neg.f64 2)) (neg.f64 beta))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 2 beta)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (pow.f64 alpha 2)) (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))))): 5 points increase in error, 4 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2)))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (neg.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 beta (*.f64 (neg.f64 -1) (+.f64 beta 2))))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (neg.f64 (+.f64 beta (*.f64 (Rewrite=> metadata-eval 1) (+.f64 beta 2)))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (neg.f64 (+.f64 beta (Rewrite=> *-lft-identity_binary64 (+.f64 beta 2)))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 beta 2) beta))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 (+.f64 beta 2)) (neg.f64 beta))) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2))) (neg.f64 beta)) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (/.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (*.f64 -1 (+.f64 beta 2)) beta)) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 -1 (+.f64 beta 2)) (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))) (/.f64 beta (/.f64 (pow.f64 alpha 2) (+.f64 beta 2))))))): 2 points increase in error, 1 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 -1 (+.f64 beta 2)) (+.f64 beta 2)) (pow.f64 alpha 2))) (/.f64 beta (/.f64 (pow.f64 alpha 2) (+.f64 beta 2)))))): 7 points increase in error, 2 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (+.f64 beta 2) (+.f64 beta 2)))) (pow.f64 alpha 2)) (/.f64 beta (/.f64 (pow.f64 alpha 2) (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (-.f64 (/.f64 (*.f64 -1 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2))) (pow.f64 alpha 2)) (/.f64 beta (/.f64 (pow.f64 alpha 2) (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)))) (/.f64 beta (/.f64 (pow.f64 alpha 2) (+.f64 beta 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (+.f64 (*.f64 2 (/.f64 1 alpha)) (-.f64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 beta (+.f64 beta 2)) (pow.f64 alpha 2)))))): 1 points increase in error, 3 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 2 (/.f64 1 alpha)) (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)))) (/.f64 (*.f64 beta (+.f64 beta 2)) (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 beta alpha) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 2 (/.f64 1 alpha)))) (/.f64 (*.f64 beta (+.f64 beta 2)) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 beta alpha)) (-.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 2 (/.f64 1 alpha))) (/.f64 (*.f64 beta (+.f64 beta 2)) (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 2 (/.f64 beta alpha)) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 2 (/.f64 1 alpha)))) (/.f64 (*.f64 beta (+.f64 beta 2)) (pow.f64 alpha 2)))): 1 points increase in error, 0 points decrease in error

   if -0.99950000000000006 < (/.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

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

   3. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha} + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{-2 - \beta \cdot 2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1732

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{-2 - \beta \cdot 2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	18.0
Cost	1236

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -8.2170100677784 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 6.441500767871081 \cdot 10^{-265}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 4.366143538667495 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \mathbf{elif}\;\alpha \leq 2.9371035026511213 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 191365.92128133672:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\alpha}{\beta + 1}}}{2}\\ \end{array} \]

Alternative 5
Error	17.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.311188220593473 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 6
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 191365.92128133672:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.311188220593473 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	18.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.311188220593473 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 9
Error	18.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.311188220593473 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022296 
(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))