Octave 3.8, jcobi/1

Average Error: 16.5 → 0.3

Time: 18.3s

Precision: binary64

Cost: 17348

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{1 + \left(1 + \left(-1 - {t_1}^{6}\right)\right)}{\left(t_1 + \left(1 + \frac{\frac{\alpha}{\frac{t_0}{\alpha}}}{t_0}\right)\right) \cdot \left(1 + {t_1}^{3}\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ alpha t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/
      (+
       (/ beta (+ beta (+ alpha 2.0)))
       (/
        (+ 1.0 (+ 1.0 (- -1.0 (pow t_1 6.0))))
        (*
         (+ t_1 (+ 1.0 (/ (/ alpha (/ t_0 alpha)) t_0)))
         (+ 1.0 (pow t_1 3.0)))))
      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 = alpha + (beta + 2.0);
	double t_1 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((1.0 + (1.0 + (-1.0 - pow(t_1, 6.0)))) / ((t_1 + (1.0 + ((alpha / (t_0 / alpha)) / t_0))) * (1.0 + pow(t_1, 3.0))))) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = alpha / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / (beta + (alpha + 2.0d0))) + ((1.0d0 + (1.0d0 + ((-1.0d0) - (t_1 ** 6.0d0)))) / ((t_1 + (1.0d0 + ((alpha / (t_0 / alpha)) / t_0))) * (1.0d0 + (t_1 ** 3.0d0))))) / 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 t_0 = alpha + (beta + 2.0);
	double t_1 = alpha / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((1.0 + (1.0 + (-1.0 - Math.pow(t_1, 6.0)))) / ((t_1 + (1.0 + ((alpha / (t_0 / alpha)) / t_0))) * (1.0 + Math.pow(t_1, 3.0))))) / 2.0;
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = alpha / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / (beta + (alpha + 2.0))) + ((1.0 + (1.0 + (-1.0 - math.pow(t_1, 6.0)))) / ((t_1 + (1.0 + ((alpha / (t_0 / alpha)) / t_0))) * (1.0 + math.pow(t_1, 3.0))))) / 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(alpha + Float64(beta + 2.0))
	t_1 = Float64(alpha / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99998)
		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))) + Float64(Float64(1.0 + Float64(1.0 + Float64(-1.0 - (t_1 ^ 6.0)))) / Float64(Float64(t_1 + Float64(1.0 + Float64(Float64(alpha / Float64(t_0 / alpha)) / t_0))) * Float64(1.0 + (t_1 ^ 3.0))))) / 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)
	t_0 = alpha + (beta + 2.0);
	t_1 = alpha / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / (beta + (alpha + 2.0))) + ((1.0 + (1.0 + (-1.0 - (t_1 ^ 6.0)))) / ((t_1 + (1.0 + ((alpha / (t_0 / alpha)) / t_0))) * (1.0 + (t_1 ^ 3.0))))) / 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_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99998], 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[(N[(1.0 + N[(1.0 + N[(-1.0 - N[Power[t$95$1, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 + N[(1.0 + N[(N[(alpha / N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[t$95$1, 3.0], $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)) < -0.99997999999999998

   1. Initial program 59.4

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

   2. Simplified 59.4

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

      [Start] 59.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.4	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.8

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

   if -0.99997999999999998 < (/.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. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.1

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} - 1}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}}{2} \]

   5. Applied egg-rr 0.1

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{\left(\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{6} + 1\right) - 1\right)} - 1}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{2} \]

   6. Applied egg-rr 0.1

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\left(\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{6} + 1\right) - 1\right) - 1}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + \color{blue}{\frac{\frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}}}{\alpha + \left(\beta + 2\right)}}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{1 + \left(1 + \left(-1 - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{6}\right)\right)}{\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(1 + \frac{\frac{\alpha}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}}}{\alpha + \left(\beta + 2\right)}\right)\right) \cdot \left(1 + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	19.3
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \frac{2}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	19.1
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \frac{2}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 5
Error	8.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9500000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2 + \frac{-4}{\alpha}}{\alpha}\\ \end{array} \]

Alternative 6
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 75000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	31.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 105000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{2}{\alpha}\\ \end{array} \]

Alternative 8
Error	40.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023059 
(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))