Octave 3.8, jcobi/1

Average Error: 16.2 → 0.3

Time: 9.4s

Precision: binary64

Cost: 8132

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.5)
   (/
    (+ (- (/ 4.0 (pow alpha 2.0))) (* 2.0 (+ (/ 1.0 alpha) (/ beta alpha))))
    2.0)
   (/ (+ (/ (- beta alpha) (+ beta (+ alpha 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) / ((alpha + beta) + 2.0)) <= -0.5) {
		tmp = (-(4.0 / pow(alpha, 2.0)) + (2.0 * ((1.0 / alpha) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 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) / ((alpha + beta) + 2.0d0)) <= (-0.5d0)) then
        tmp = (-(4.0d0 / (alpha ** 2.0d0)) + (2.0d0 * ((1.0d0 / alpha) + (beta / alpha)))) / 2.0d0
    else
        tmp = (((beta - alpha) / (beta + (alpha + 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) / ((alpha + beta) + 2.0)) <= -0.5) {
		tmp = (-(4.0 / Math.pow(alpha, 2.0)) + (2.0 * ((1.0 / alpha) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 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) / ((alpha + beta) + 2.0)) <= -0.5:
		tmp = (-(4.0 / math.pow(alpha, 2.0)) + (2.0 * ((1.0 / alpha) + (beta / alpha)))) / 2.0
	else:
		tmp = (((beta - alpha) / (beta + (alpha + 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(alpha + beta) + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(-Float64(4.0 / (alpha ^ 2.0))) + Float64(2.0 * Float64(Float64(1.0 / alpha) + Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 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) / ((alpha + beta) + 2.0)) <= -0.5)
		tmp = (-(4.0 / (alpha ^ 2.0)) + (2.0 * ((1.0 / alpha) + (beta / alpha)))) / 2.0;
	else
		tmp = (((beta - alpha) / (beta + (alpha + 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[((-N[(4.0 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]) + N[(2.0 * N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 58.3

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

   2. Simplified 58.3

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

      [Start] 58.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational_best.json-simplify-1 [=>] 58.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
      rational_best.json-simplify-43 [=>] 58.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} + 1}{2} \]

   3. Taylor expanded in alpha around inf 4.0

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

   4. Simplified 4.0

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

      [Start] 4.0	\[ \frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(2 \cdot \frac{\beta}{\alpha} + \color{blue}{\left(2 \cdot \frac{1}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right)}\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-43 [=>] 4.0	\[ \frac{\color{blue}{\left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + \left(2 \cdot \frac{1}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right)\right)} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-2 [=>] 4.0	\[ \frac{\left(\color{blue}{\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} \cdot -1} + \left(2 \cdot \frac{1}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-12 [=>] 4.0	\[ \frac{\left(\color{blue}{\left(-\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right)} + \left(2 \cdot \frac{1}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(\left(-\frac{{\color{blue}{\left(2 + \beta\right)}}^{2}}{{\alpha}^{2}}\right) + \left(2 \cdot \frac{1}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(\left(-\frac{{\left(2 + \beta\right)}^{2}}{{\alpha}^{2}}\right) + \color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(\left(-\frac{{\left(2 + \beta\right)}^{2}}{{\alpha}^{2}}\right) + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}}{2} \]

   5. Taylor expanded in alpha around 0 4.0

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

   6. Simplified 4.0

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

      [Start] 4.0	\[ \frac{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 4.0	\[ \frac{\color{blue}{\frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} \cdot -1} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-12 [=>] 4.0	\[ \frac{\color{blue}{\left(-\frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(-\frac{\color{blue}{\beta \cdot \left(\beta + 2\right) + {\left(\beta + 2\right)}^{2}}}{{\alpha}^{2}}\right) + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(-\frac{\beta \cdot \color{blue}{\left(2 + \beta\right)} + {\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right) + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-1 [=>] 4.0	\[ \frac{\left(-\frac{\beta \cdot \left(2 + \beta\right) + {\color{blue}{\left(2 + \beta\right)}}^{2}}{{\alpha}^{2}}\right) + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      rational_best.json-simplify-47 [=>] 4.0	\[ \frac{\left(-\frac{\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}}{{\alpha}^{2}}\right) + \color{blue}{2 \cdot \left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right)}}{2} \]

   7. Taylor expanded in beta around 0 1.1

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

   if -0.5 < (/.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

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

      [Start] 0.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
      rational_best.json-simplify-43 [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} + 1}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\left(-\frac{4}{{\alpha}^{2}}\right) + 2 \cdot \left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	1476

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

Alternative 2
Error	20.7
Cost	848

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.15 \cdot 10^{-223}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -9.5 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 430000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	20.7
Cost	848

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \beta + 1}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.2 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -6.2 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 430000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	8.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.16 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	18.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 430000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	32.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023089 
(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))