Octave 3.8, jcobi/1

Average Accuracy: 74.5% → 99.9%

Time: 18.9s

Precision: binary64

Cost: 1728

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

Math

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

↓

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

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 (+ 2.0 beta))))
   (/ (/ 1.0 (+ (+ (/ 2.0 t_0) (/ beta t_0)) (/ alpha t_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 + (2.0 + beta);
	return (1.0 / (((2.0 / t_0) + (beta / t_0)) + (alpha / t_0))) / 2.0;
}

Fortran

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

↓

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

Java

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

↓

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

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

↓

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

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(beta + Float64(2.0 + beta))
	return Float64(Float64(1.0 / Float64(Float64(Float64(2.0 / t_0) + Float64(beta / t_0)) + Float64(alpha / t_0))) / 2.0)
end

MATLAB

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

↓

function tmp = code(alpha, beta)
	t_0 = beta + (2.0 + beta);
	tmp = (1.0 / (((2.0 / t_0) + (beta / t_0)) + (alpha / t_0))) / 2.0;
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[(beta + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[(N[(N[(2.0 / t$95$0), $MachinePrecision] + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Initial program 74.5%

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

2. Simplified 74.5%

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

   [Start] 74.5	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   +-commutative [=>] 74.5	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

3. Applied egg-rr 48.8%

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

   [Start] 74.5	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
   +-commutative [=>] 74.5	\[ \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}{2} \]
   flip-+ [=>] 48.8	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}{1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}}{2} \]
   metadata-eval [<=] 48.8	\[ \frac{\frac{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}{\color{blue}{1 \cdot 1} - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}{2} \]
   *-rgt-identity [<=] 48.8	\[ \frac{\frac{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}{1 \cdot 1 - \color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1}}}{2} \]
   metadata-eval [=>] 48.8	\[ \frac{\frac{\color{blue}{1} - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1}}{2} \]
   pow2 [=>] 48.8	\[ \frac{\frac{1 - \color{blue}{{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{2}}}{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1}}{2} \]
   associate-+l+ [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}^{2}}{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1}}{2} \]
   metadata-eval [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}{\color{blue}{1} - \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot 1}}{2} \]
   *-rgt-identity [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}{1 - \color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}}{2} \]
   associate-+l+ [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}{1 - \frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}}{2} \]

4. Simplified 48.8%

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

   [Start] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
   +-commutative [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
   +-commutative [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
   +-commutative [<=] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
   +-commutative [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}}}{2} \]
   +-commutative [=>] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}}}{2} \]
   +-commutative [<=] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}}}{2} \]

5. Applied egg-rr 74.5%

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

   [Start] 48.8	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}}{2} \]
   metadata-eval [<=] 48.8	\[ \frac{\frac{\color{blue}{1 \cdot 1} - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}}{2} \]
   unpow2 [=>] 48.8	\[ \frac{\frac{1 \cdot 1 - \color{blue}{\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta} \cdot \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}}{1 - \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}}{2} \]
   flip-+ [<=] 74.5	\[ \frac{\color{blue}{1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}}{2} \]
   add-log-exp [=>] 74.5	\[ \frac{\color{blue}{\log \left(e^{1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}\right)}}{2} \]
   metadata-eval [<=] 74.5	\[ \frac{\log \left(e^{\color{blue}{\left(1 - 0\right)} + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}\right)}{2} \]
   metadata-eval [<=] 74.5	\[ \frac{\log \left(e^{\left(1 - \color{blue}{\log 1}\right) + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}}\right)}{2} \]
   associate--r- [<=] 74.5	\[ \frac{\log \left(e^{\color{blue}{1 - \left(\log 1 - \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}\right)}{2} \]
   metadata-eval [=>] 74.5	\[ \frac{\log \left(e^{1 - \left(\color{blue}{0} - \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}\right)}{2} \]
   neg-sub0 [<=] 74.5	\[ \frac{\log \left(e^{1 - \color{blue}{\left(-\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}\right)}{2} \]
   frac-2neg [=>] 74.5	\[ \frac{\log \left(e^{1 - \left(-\color{blue}{\frac{-\left(\beta - \alpha\right)}{-\left(\left(\alpha + 2\right) + \beta\right)}}\right)}\right)}{2} \]
   distribute-neg-frac [=>] 74.5	\[ \frac{\log \left(e^{1 - \color{blue}{\frac{-\left(-\left(\beta - \alpha\right)\right)}{-\left(\left(\alpha + 2\right) + \beta\right)}}}\right)}{2} \]

6. Applied egg-rr 74.5%

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

   [Start] 74.5	\[ \frac{\log \left(e^{1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}\right)}{2} \]
   add-log-exp [<=] 74.5	\[ \frac{\color{blue}{1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}{2} \]
   flip-- [=>] 48.8	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}{1 + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}}{2} \]
   clear-num [=>] 48.8	\[ \frac{\color{blue}{\frac{1}{\frac{1 + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}{1 \cdot 1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}}}{2} \]
   *-un-lft-identity [=>] 48.8	\[ \frac{\frac{1}{\frac{\color{blue}{1 \cdot \left(1 + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}\right)}}{1 \cdot 1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}}{2} \]
   associate-/l* [=>] 48.8	\[ \frac{\frac{1}{\color{blue}{\frac{1}{\frac{1 \cdot 1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}{1 + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}}}}{2} \]
   flip-- [<=] 74.5	\[ \frac{\frac{1}{\frac{1}{\color{blue}{1 - \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}}}}}{2} \]

7. Taylor expanded in alpha around inf 99.9%

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

8. Simplified 99.9%

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

   [Start] 99.9	\[ \frac{\frac{1}{2 \cdot \frac{1}{\beta - -1 \cdot \left(\beta + 2\right)} + \left(\frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right)}}{2} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{1}{2 \cdot \frac{1}{\beta - -1 \cdot \left(\beta + 2\right)} + \color{blue}{\left(\frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)} + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}\right)}}}{2} \]
   associate-+r+ [=>] 99.9	\[ \frac{\frac{1}{\color{blue}{\left(2 \cdot \frac{1}{\beta - -1 \cdot \left(\beta + 2\right)} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}}{2} \]
   associate-*r/ [=>] 99.9	\[ \frac{\frac{1}{\left(\color{blue}{\frac{2 \cdot 1}{\beta - -1 \cdot \left(\beta + 2\right)}} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{\color{blue}{2}}{\beta - -1 \cdot \left(\beta + 2\right)} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   sub-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\color{blue}{\beta + \left(--1 \cdot \left(\beta + 2\right)\right)}} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   mul-1-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \left(-\color{blue}{\left(-\left(\beta + 2\right)\right)}\right)} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   remove-double-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \color{blue}{\left(\beta + 2\right)}} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \color{blue}{\left(2 + \beta\right)}} + \frac{\beta}{\beta - -1 \cdot \left(\beta + 2\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   sub-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \left(2 + \beta\right)} + \frac{\beta}{\color{blue}{\beta + \left(--1 \cdot \left(\beta + 2\right)\right)}}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   mul-1-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \left(2 + \beta\right)} + \frac{\beta}{\beta + \left(-\color{blue}{\left(-\left(\beta + 2\right)\right)}\right)}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   remove-double-neg [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \left(2 + \beta\right)} + \frac{\beta}{\beta + \color{blue}{\left(\beta + 2\right)}}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{1}{\left(\frac{2}{\beta + \left(2 + \beta\right)} + \frac{\beta}{\beta + \color{blue}{\left(2 + \beta\right)}}\right) + \frac{\alpha}{\beta - -1 \cdot \left(\beta + 2\right)}}}{2} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1476

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

Alternative 2
Accuracy	92.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Accuracy	70.7%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.3 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	70.9%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.3 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 5
Accuracy	87.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	92.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	70.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.25 \cdot 10^{-103}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -2.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 64:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	71.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 64:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	49.9%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023138 
(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))