Octave 3.8, jcobi/1

Average Error: 16.5 → 0.3

Time: 13.4s

Precision: binary64

Cost: 14660

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]

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)) -0.9995)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
   (/
    (-
     (/ beta (+ beta (+ alpha 2.0)))
     (log (exp (+ (/ alpha (+ alpha (+ beta 2.0))) -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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - log(exp(((alpha / (alpha + (beta + 2.0))) + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9995d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / (beta + (alpha + 2.0d0))) - log(exp(((alpha / (alpha + (beta + 2.0d0))) + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

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)) <= -0.9995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) - Math.log(Math.exp(((alpha / (alpha + (beta + 2.0))) + -1.0)))) / 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)) <= -0.9995:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) - math.log(math.exp(((alpha / (alpha + (beta + 2.0))) + -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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) - log(exp(Float64(Float64(alpha / Float64(alpha + Float64(beta + 2.0))) + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

↓

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9995)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / (beta + (alpha + 2.0))) - log(exp(((alpha / (alpha + (beta + 2.0))) + -1.0)))) / 2.0;
	end
	tmp_2 = 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], -0.9995], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[Exp[N[(N[(alpha / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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)) < -0.99950000000000006

   1. Initial program 59.3

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

   2. Taylor expanded in alpha around inf 0.9

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

   if -0.99950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\log \left(e^{\frac{\alpha}{\alpha + \left(2 + \beta\right)} + -1}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + -1}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	3012

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{1 - \frac{\frac{\alpha}{t_0}}{\frac{t_0}{\alpha}}}{\frac{\alpha}{\alpha + \left(\beta + 2\right)} + 1}}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	20.6
Cost	1372

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.52 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -5.5 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.6 \cdot 10^{-306}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 5.4 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 0.0062:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	20.5
Cost	1372

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -2.3 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -3.2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.6 \cdot 10^{-306}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 0.0062:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 6
Error	20.5
Cost	1112

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.52 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -5 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.6 \cdot 10^{-306}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{-131}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	7.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.9:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.9:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	18.8
Cost	196

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

Alternative 10
Error	32.5
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022325 
(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))