Octave 3.8, jcobi/1

Average Accuracy: 74.9% → 99.7%

Time: 8.1s

Precision: binary64

Cost: 8388

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

Math

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

↓

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

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)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ (/ beta 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 = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / 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(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / 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[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / 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 8.5%

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

   2. Simplified 8.5%

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

      [Start] 8.5	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 8.5	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 93.9%

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

   4. Simplified 99.0%

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

      [Start] 93.9	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      +-commutative [=>] 93.9	\[ \frac{\color{blue}{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + -1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}}}{2} \]
      mul-1-neg [=>] 93.9	\[ \frac{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \color{blue}{\left(-\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}\right)}}{2} \]
      unsub-neg [=>] 93.9	\[ \frac{\color{blue}{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} - \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}}}{2} \]

   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%

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

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1860

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

Alternative 2
Accuracy	99.4%
Cost	1476

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

Alternative 3
Accuracy	87.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3800000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Accuracy	93.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3800000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	69.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.35:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	68.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	71.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	49.9%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023152 
(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))