Octave 3.8, jcobi/1

Average Accuracy: 75.2% → 99.7%

Time: 10.9s

Precision: binary64

Cost: 8452

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

Math

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

↓

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.6%

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

   2. Simplified 8.6%

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

      [Start] 8.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 8.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 94.4%

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

   4. Simplified 94.4%

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

      [Start] 94.4	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 94.4	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Applied egg-rr 94.4%

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

   6. Simplified 98.9%

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

      [Start] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \left(-\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right) \cdot \frac{1}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      associate-*r/ [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{\left(-\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)\right) \cdot 1}{\alpha \cdot \left(-\alpha\right)}}\right)}{2} \]
      *-rgt-identity [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{-\left(\beta + 2\right) \cdot \left(\beta + \left(\beta + 2\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      distribute-lft-neg-in [=>] 94.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(-\left(\beta + 2\right)\right) \cdot \left(\beta + \left(\beta + 2\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      times-frac [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{-\left(\beta + 2\right)}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{-\alpha}}\right)}{2} \]
      neg-sub0 [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{0 - \left(\beta + 2\right)}}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{-\alpha}\right)}{2} \]
      +-commutative [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{0 - \color{blue}{\left(2 + \beta\right)}}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{-\alpha}\right)}{2} \]
      associate--r+ [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(0 - 2\right) - \beta}}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{-\alpha}\right)}{2} \]
      metadata-eval [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{-2} - \beta}{\alpha} \cdot \frac{\beta + \left(\beta + 2\right)}{-\alpha}\right)}{2} \]
      associate-+r+ [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{-2 - \beta}{\alpha} \cdot \frac{\color{blue}{\left(\beta + \beta\right) + 2}}{-\alpha}\right)}{2} \]
      count-2 [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{-2 - \beta}{\alpha} \cdot \frac{\color{blue}{2 \cdot \beta} + 2}{-\alpha}\right)}{2} \]
      fma-def [=>] 98.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{-2 - \beta}{\alpha} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{-\alpha}\right)}{2} \]

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

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1476

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

Alternative 2
Accuracy	68.3%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 1.26 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	68.5%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 4
Accuracy	87.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2150000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	93.3%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 175000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	67.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{-217}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	71.6%
Cost	196

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

Alternative 8
Accuracy	49.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023126 
(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))