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

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

↓

\[\begin{array}{l} t_0 := \frac{\beta + 2}{\alpha}\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\frac{\beta}{t_1} - \left(t_0 \cdot \left(t_0 + -1\right) - {t_0}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t_1}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ beta 2.0) alpha)) (t_1 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
     (/ (- (/ beta t_1) (- (* t_0 (+ t_0 -1.0)) (pow t_0 3.0))) 2.0)
     (/ (fma (/ 1.0 t_1) (- beta alpha) 1.0) 2.0))))

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 + 2.0) / alpha;
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = ((beta / t_1) - ((t_0 * (t_0 + -1.0)) - pow(t_0, 3.0))) / 2.0;
	} else {
		tmp = fma((1.0 / t_1), (beta - alpha), 1.0) / 2.0;
	}
	return tmp;
}

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 + 2.0) / alpha)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(Float64(Float64(beta / t_1) - Float64(Float64(t_0 * Float64(t_0 + -1.0)) - (t_0 ^ 3.0))) / 2.0);
	else
		tmp = Float64(fma(Float64(1.0 / t_1), Float64(beta - alpha), 1.0) / 2.0);
	end
	return tmp
end

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 + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(N[(N[(beta / t$95$1), $MachinePrecision] - N[(N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(beta - alpha), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

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

↓

\begin{array}{l}
t_0 := \frac{\beta + 2}{\alpha}\\
t_1 := \beta + \left(\alpha + 2\right)\\
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\
\;\;\;\;\frac{\frac{\beta}{t_1} - \left(t_0 \cdot \left(t_0 + -1\right) - {t_0}^{3}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t_1}, \beta - \alpha, 1\right)}{2}\\


\end{array}

Derivation

Split input into 2 regimes
if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.98999999999999999
1. Initial program 58.9
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. Applied egg-rr57.0
  \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}}{2} \]
3. Taylor expanded in alpha around inf 5.5
  \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(-1 \cdot \frac{\beta + 2}{\alpha} + \left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)\right)}}{2} \]
4. Simplified0.2
  \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\frac{\beta + 2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right) - {\left(\frac{\beta + 2}{\alpha}\right)}^{3}\right)}}{2} \]
  Proof
  (-.f64 (*.f64 (/.f64 (+.f64 beta 2) alpha) (+.f64 -1 (/.f64 (+.f64 beta 2) alpha))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
  (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 beta 2) alpha)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)) (*.f64 alpha alpha)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 7 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha alpha))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 (+.f64 beta 2) alpha) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 beta 2) alpha))))): 5 points increase in error, 7 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)) (*.f64 alpha alpha))))): 5 points increase in error, 18 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha alpha)))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (pow.f64 (+.f64 beta 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha (pow.f64 alpha 2))))): 9 points increase in error, 6 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (*.f64 (+.f64 beta 2) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)))) (*.f64 alpha (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 (+.f64 beta 2) 3)) (*.f64 alpha (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (pow.f64 (+.f64 beta 2) 3) (*.f64 alpha (Rewrite=> unpow2_binary64 (*.f64 alpha alpha))))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (pow.f64 (+.f64 beta 2) 3) (Rewrite<= cube-mult_binary64 (pow.f64 alpha 3)))): 7 points increase in error, 4 points decrease in error
  (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (neg.f64 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3))))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (+.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3)))))): 0 points increase in error, 0 points decrease in error
if -0.98999999999999999 < (/.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-rr0.0
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}}{2} \]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\beta + 2}{\alpha} \cdot \left(\frac{\beta + 2}{\alpha} + -1\right) - {\left(\frac{\beta + 2}{\alpha}\right)}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \left(\alpha + 2\right)}, \beta - \alpha, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1988

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

Alternative 2
Error	0.4
Cost	1860

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

Alternative 3
Error	0.4
Cost	1476

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

Alternative 4
Error	4.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.622656214044243 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 115749.45209133253:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1917497187181874.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 5
Error	17.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.7435611973324852 \cdot 10^{-77}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1917497187181874.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 6
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1917497187181874.8:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 7
Error	17.5
Cost	584

Alternative 8
Error	21.0
Cost	456

Alternative 9
Error	46.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 21624698785539.45:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 10
Error	20.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 0.0009646464963126234:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 11
Error	58.4
Cost	192

\[\frac{\beta}{\alpha} \]

Error

Derivation

`if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.98999999999999999`

`if -0.98999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))`

Alternatives

Error

Reproduce