Octave 3.8, jcobi/3

Average Error: 93.8% → 99.6%

Time: 40.8s

Precision: binary64

Cost: 1732.00

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{3 + \left(\beta + \alpha\right)} \cdot \left(\beta + 1\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 2.2e+25)
     (/ (* (/ (+ alpha 1.0) (+ 3.0 (+ beta alpha))) (+ beta 1.0)) (* t_0 t_0))
     (/
      (/ (+ alpha 1.0) (+ (+ beta 3.0) (* alpha 2.0)))
      (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.2e+25) {
		tmp = (((alpha + 1.0) / (3.0 + (beta + alpha))) * (beta + 1.0)) / (t_0 * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 2.2d+25) then
        tmp = (((alpha + 1.0d0) / (3.0d0 + (beta + alpha))) * (beta + 1.0d0)) / (t_0 * t_0)
    else
        tmp = ((alpha + 1.0d0) / ((beta + 3.0d0) + (alpha * 2.0d0))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2.2e+25) {
		tmp = (((alpha + 1.0) / (3.0 + (beta + alpha))) * (beta + 1.0)) / (t_0 * t_0);
	} else {
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 2.0))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2.2e+25:
		tmp = (((alpha + 1.0) / (3.0 + (beta + alpha))) * (beta + 1.0)) / (t_0 * t_0)
	else:
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 2.0))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2.2e+25)
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(3.0 + Float64(beta + alpha))) * Float64(beta + 1.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(beta + 3.0) + Float64(alpha * 2.0))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2.2e+25)
		tmp = (((alpha + 1.0) / (3.0 + (beta + alpha))) * (beta + 1.0)) / (t_0 * t_0);
	else
		tmp = ((alpha + 1.0) / ((beta + 3.0) + (alpha * 2.0))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2e+25], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2000000000000001e25

   1. Initial program 99.9

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

   2. Simplified 99.9

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

      [Start] 99.9	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 99.9	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/r* [<=] 99.9	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
      +-commutative [=>] 99.9	\[ \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      associate-+l+ [=>] 99.9	\[ \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      associate-+r+ [=>] 99.9	\[ \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      *-lft-identity [<=] 99.9	\[ \frac{\left(1 + \alpha\right) + \left(\color{blue}{1 \cdot \beta} + \beta \cdot \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{\left(1 + \alpha\right) + \left(1 \cdot \beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      distribute-rgt-in [<=] 99.9	\[ \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      distribute-rgt1-in [=>] 99.9	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]
      *-commutative [=>] 99.8	\[ \color{blue}{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   if 2.2000000000000001e25 < beta

   1. Initial program 88.4

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

   2. Simplified 99.8

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

      [Start] 88.4	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Taylor expanded in beta around inf 99.4

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

   4. Simplified 99.4

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

      [Start] 99.4	\[ \frac{\frac{\alpha + 1}{\beta + \left(3 + 2 \cdot \alpha\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate-+r+ [=>] 99.4	\[ \frac{\frac{\alpha + 1}{\color{blue}{\left(\beta + 3\right) + 2 \cdot \alpha}}}{\alpha + \left(\beta + 3\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{3 + \left(\beta + \alpha\right)} \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.8%
Cost	1600.00

\[\begin{array}{l} t_0 := \left(-2 - \alpha\right) - \beta\\ \frac{\frac{\frac{-1 - \beta}{t_0}}{\frac{t_0}{-1 - \alpha}}}{\alpha + \left(\beta + 3\right)} \end{array} \]

Alternative 2
Error	99.8%
Cost	1600.00

\[\begin{array}{l} t_0 := -2 - \left(\beta + \alpha\right)\\ \frac{\frac{-1 - \beta}{t_0} \cdot \frac{-1 - \alpha}{t_0}}{\alpha + \left(\beta + 3\right)} \end{array} \]

Alternative 3
Error	98.7%
Cost	1348.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 43000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 4
Error	98.7%
Cost	1348.00

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 45000000:\\ \;\;\;\;\frac{\frac{\frac{-1 - \beta}{\left(-2 - \alpha\right) - \beta}}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 5
Error	98.4%
Cost	1220.00

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 0.62:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta + 2}}{2 - \beta}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 6
Error	97.8%
Cost	1092.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.62:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(2 - \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1}{\beta + 3}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	98.2%
Cost	1092.00

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 0.65:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta + 2}}{2 - \beta}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1}{\beta + 3}}{t_0}\\ \end{array} \]

Alternative 8
Error	97.3%
Cost	964.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{-0.5}{\beta + 2} \cdot \frac{1}{-3 - \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Error	97.2%
Cost	964.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(2 - \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	96.9%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 11
Error	93.7%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Error	96.8%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.65:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	45.9%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 14
Error	91.1%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	44.3%
Cost	64.00

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023093 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))