Octave 3.8, jcobi/1

Average Error: 16.5 → 0.4

Time: 14.0s

Precision: binary64

Cost: 14148

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -1.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ 2.0 alpha))))) 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) / (2.0 + (beta + alpha))) <= -1.0) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (2.0 + alpha))))) / 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) / (2.0 + (beta + alpha))) <= -1.0) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (2.0 + alpha))))) / 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) / (2.0 + (beta + alpha))) <= -1.0:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (2.0 + alpha))))) / 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(2.0 + Float64(beta + alpha))) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(2.0 + alpha))))) / 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $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)) < -1

   1. Initial program 60.5

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

   2. Simplified 60.5

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 beta alpha) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2)) 1) 2): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in alpha around inf 0.0

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.5

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 beta alpha) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2)) 1) 2): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.5

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

   4. Simplified 0.5

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]
      Proof
      (/.f64 (exp.f64 (log1p.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha 2) beta)))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (log1p.f64 (/.f64 (-.f64 beta alpha) (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 2 alpha)) beta)))) 2): 4 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (log1p.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= +-commutative_binary64 (+.f64 beta (+.f64 2 alpha)))))) 2): 0 points increase in error, 4 points decrease in error
      (/.f64 (exp.f64 (log1p.f64 (/.f64 (-.f64 beta alpha) (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 alpha 2)))))) 2): 0 points increase in error, 4 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	15168

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

Alternative 2
Error	0.4
Cost	1476

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

Alternative 3
Error	16.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(1 + \beta\right)}{2}\\ \end{array} \]

Alternative 4
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 350000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(1 + \beta\right)}{2}\\ \end{array} \]

Alternative 5
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6200000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	19.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.2:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	19.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.8:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	18.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 21000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	32.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022349 
(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))