Octave 3.8, jcobi/1

Average Accuracy: 74.6% → 99.8%

Time: 12.4s

Precision: binary64

Cost: 14148

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/ (+ (/ 2.0 alpha) (- (* 2.0 (/ beta alpha)) (/ 4.0 (* alpha alpha)))) 2.0)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((2.0 / alpha) + ((2.0 * (beta / alpha)) - (4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((2.0 / alpha) + ((2.0 * (beta / alpha)) - (4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995:
		tmp = ((2.0 / alpha) + ((2.0 * (beta / alpha)) - (4.0 / (alpha * alpha)))) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(2.0 / alpha) + Float64(Float64(2.0 * Float64(beta / alpha)) - Float64(4.0 / Float64(alpha * alpha)))) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99995], N[(N[(N[(2.0 / alpha), $MachinePrecision] + N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(4.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999950000000000006

   1. Initial program 7.7%

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

   2. Simplified 7.7%

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

      [Start] 7.7	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 7.7	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 94.9%

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

   4. Simplified 94.9%

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

      [Start] 94.9	\[ \frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      fma-def [=>] 94.9	\[ \frac{\color{blue}{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, -1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      fma-def [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \color{blue}{\mathsf{fma}\left(-1, \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}, 2 \cdot \frac{1}{\alpha}\right)}\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      +-commutative [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\color{blue}{\left(2 + \beta\right)}}^{2}}{{\alpha}^{2}}, 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      unpow2 [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\color{blue}{\alpha \cdot \alpha}}, 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      associate-*r/ [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      metadata-eval [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{\color{blue}{2}}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      associate-/l* [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \color{blue}{\frac{\beta}{\frac{{\alpha}^{2}}{\beta + 2}}}}{2} \]
      unpow2 [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \frac{\beta}{\frac{\color{blue}{\alpha \cdot \alpha}}{\beta + 2}}}{2} \]
      +-commutative [=>] 94.9	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \frac{\beta}{\frac{\alpha \cdot \alpha}{\color{blue}{2 + \beta}}}}{2} \]

   5. Taylor expanded in beta around 0 99.2%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + \beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2} \]

   6. Simplified 99.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right)}}{2} \]
      Proof

      [Start] 99.2	\[ \frac{\left(2 \cdot \frac{1}{\alpha} + \beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]
      associate--l+ [=>] 99.2	\[ \frac{\color{blue}{2 \cdot \frac{1}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2} \]
      associate-*r/ [=>] 99.2	\[ \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 99.2	\[ \frac{\frac{\color{blue}{2}}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\color{blue}{\frac{2 \cdot 1}{\alpha}} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{\color{blue}{2}}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \color{blue}{\frac{6 \cdot 1}{{\alpha}^{2}}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{\color{blue}{6}}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\color{blue}{\alpha \cdot \alpha}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}\right)}{2} \]
      metadata-eval [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{\color{blue}{4}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 99.2	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   7. Taylor expanded in alpha around inf 99.3%

      \[\leadsto \frac{\frac{2}{\alpha} + \left(\color{blue}{2 \cdot \frac{\beta}{\alpha}} - \frac{4}{\alpha \cdot \alpha}\right)}{2} \]

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
      Proof

      [Start] 99.9	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      add-exp-log [=>] 99.9	\[ \frac{\color{blue}{e^{\log \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{e^{\log \color{blue}{\left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]
      log1p-def [=>] 99.9	\[ \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]
      associate-+l+ [=>] 99.9	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}}{2} \]

   4. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]
      Proof

      [Start] 99.9	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}\right)}}{2} \]
      +-commutative [=>] 99.9	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]
      +-commutative [<=] 99.9	\[ \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1732

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Accuracy	87.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 335:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Accuracy	92.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 335:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	69.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{-306}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	70.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{-306}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 1.95 \cdot 10^{-262}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	71.1%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	48.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023125 
(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))