Octave 3.8, jcobi/1

Average Accuracy: 75.2% → 99.6%

Time: 11.1s

Precision: binary64

Cost: 8260

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(\frac{\beta - -2}{\alpha} - \frac{\frac{\beta + 2}{\alpha}}{\frac{\alpha}{\beta + 2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{1}{t_0}, 1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
     (/
      (+
       (/ beta t_0)
       (-
        (/ (- beta -2.0) alpha)
        (/ (/ (+ beta 2.0) alpha) (/ alpha (+ beta 2.0)))))
      2.0)
     (/ (fma beta (/ 1.0 t_0) (- 1.0 (/ 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 + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = ((beta / t_0) + (((beta - -2.0) / alpha) - (((beta + 2.0) / alpha) / (alpha / (beta + 2.0))))) / 2.0;
	} else {
		tmp = fma(beta, (1.0 / t_0), (1.0 - (alpha / t_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(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(Float64(beta - -2.0) / alpha) - Float64(Float64(Float64(beta + 2.0) / alpha) / Float64(alpha / Float64(beta + 2.0))))) / 2.0);
	else
		tmp = Float64(fma(beta, Float64(1.0 / t_0), Float64(1.0 - Float64(alpha / t_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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta * N[(1.0 / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 9.2%

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

   2. Simplified 9.2%

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

      [Start] 9.2	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 9.2	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 12.0%

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

      [Start] 9.2	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      div-sub [=>] 9.2	\[ \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]
      associate-+l- [=>] 12.0	\[ \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]
      associate-+l+ [=>] 12.0	\[ \frac{\frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]
      associate-+l+ [=>] 12.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)}{2} \]

   4. Taylor expanded in alpha around inf 91.1%

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(-1 \cdot \frac{\beta + 2}{\alpha} + \left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)\right)}}{2} \]

   5. Simplified 94.9%

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - {\left(\frac{\beta + 2}{\alpha}\right)}^{3}\right)}}{2} \]
      Proof

      [Start] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(-1 \cdot \frac{\beta + 2}{\alpha} + \left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)\right)}{2} \]
      associate-+r+ [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\left(-1 \cdot \frac{\beta + 2}{\alpha} + \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right) + -1 \cdot \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}}{2} \]
      mul-1-neg [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(-1 \cdot \frac{\beta + 2}{\alpha} + \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right) + \color{blue}{\left(-\frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}\right)}{2} \]
      unsub-neg [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\left(-1 \cdot \frac{\beta + 2}{\alpha} + \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right) - \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}}{2} \]
      +-commutative [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{\left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{\beta + 2}{\alpha}\right)} - \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}{2} \]
      mul-1-neg [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + \color{blue}{\left(-\frac{\beta + 2}{\alpha}\right)}\right) - \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}{2} \]
      unsub-neg [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{\left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} - \frac{\beta + 2}{\alpha}\right)} - \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}{2} \]
      unpow2 [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\color{blue}{\alpha \cdot \alpha}} - \frac{\beta + 2}{\alpha}\right) - \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)}{2} \]
      cube-mult [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 2\right)\right)}}{{\alpha}^{3}}\right)}{2} \]
      unpow2 [<=] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\left(\beta + 2\right) \cdot \color{blue}{{\left(\beta + 2\right)}^{2}}}{{\alpha}^{3}}\right)}{2} \]
      cube-mult [=>] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\left(\beta + 2\right) \cdot {\left(\beta + 2\right)}^{2}}{\color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}\right)}{2} \]
      unpow2 [<=] 91.1	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\left(\beta + 2\right) \cdot {\left(\beta + 2\right)}^{2}}{\alpha \cdot \color{blue}{{\alpha}^{2}}}\right)}{2} \]
      times-frac [=>] 94.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \color{blue}{\frac{\beta + 2}{\alpha} \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}}\right)}{2} \]
      unpow2 [=>] 94.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\beta + 2}{\alpha} \cdot \frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 94.9	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right) - \frac{\beta + 2}{\alpha} \cdot \frac{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   6. Taylor expanded in alpha around inf 94.5%

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

   7. Simplified 98.7%

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

      [Start] 94.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(-1 \cdot \frac{\beta + 2}{\alpha} + \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 94.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}}{2} \]
      unpow2 [=>] 94.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{{\left(\beta + 2\right)}^{2}}{\color{blue}{\alpha \cdot \alpha}} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      unpow2 [=>] 94.5	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\color{blue}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{\alpha \cdot \alpha} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      times-frac [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{\frac{\beta + 2}{\alpha} \cdot \frac{\beta + 2}{\alpha}} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      unpow2 [<=] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{{\left(\frac{\beta + 2}{\alpha}\right)}^{2}} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      +-commutative [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{\color{blue}{2 + \beta}}{\alpha}\right)}^{2} + -1 \cdot \frac{\beta + 2}{\alpha}\right)}{2} \]
      mul-1-neg [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \color{blue}{\left(-\frac{\beta + 2}{\alpha}\right)}\right)}{2} \]
      distribute-neg-frac [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \color{blue}{\frac{-\left(\beta + 2\right)}{\alpha}}\right)}{2} \]
      +-commutative [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \frac{-\color{blue}{\left(2 + \beta\right)}}{\alpha}\right)}{2} \]
      distribute-neg-in [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \frac{\color{blue}{\left(-2\right) + \left(-\beta\right)}}{\alpha}\right)}{2} \]
      metadata-eval [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \frac{\color{blue}{-2} + \left(-\beta\right)}{\alpha}\right)}{2} \]
      unsub-neg [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \frac{\color{blue}{-2 - \beta}}{\alpha}\right)}{2} \]

   8. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left({\left(\frac{2 + \beta}{\alpha}\right)}^{2} + \frac{-2 - \beta}{\alpha}\right)}{2} \]
      unpow2 [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{\frac{2 + \beta}{\alpha} \cdot \frac{2 + \beta}{\alpha}} + \frac{-2 - \beta}{\alpha}\right)}{2} \]
      clear-num [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{2 + \beta}{\alpha} \cdot \color{blue}{\frac{1}{\frac{\alpha}{2 + \beta}}} + \frac{-2 - \beta}{\alpha}\right)}{2} \]
      un-div-inv [=>] 98.7	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\color{blue}{\frac{\frac{2 + \beta}{\alpha}}{\frac{\alpha}{2 + \beta}}} + \frac{-2 - \beta}{\alpha}\right)}{2} \]

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 100.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      div-sub [=>] 100.0	\[ \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]
      associate-+l- [=>] 100.0	\[ \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]
      div-inv [=>] 100.0	\[ \frac{\color{blue}{\beta \cdot \frac{1}{\left(\beta + \alpha\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]
      fma-neg [=>] 100.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\beta, \frac{1}{\left(\beta + \alpha\right) + 2}, -\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)}}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\mathsf{fma}\left(\beta, \frac{1}{\color{blue}{\beta + \left(\alpha + 2\right)}}, -\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\mathsf{fma}\left(\beta, \frac{1}{\beta + \left(\alpha + 2\right)}, -\left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	2372

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{t_1 + \left(\frac{\beta - -2}{\alpha} - \frac{\frac{\beta + 2}{\alpha}}{\frac{\alpha}{\beta + 2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	1476

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 4
Accuracy	71.0%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -8.6 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -8 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	71.2%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -8.5 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -7.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 6
Accuracy	87.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 380000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	93.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3400000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	70.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -8.6 \cdot 10^{-115}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -6 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	71.7%
Cost	196

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

Alternative 10
Accuracy	49.7%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023153 
(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))