Octave 3.8, jcobi/3

Average Accuracy: 94.6% → 99.6%

Time: 21.5s

Precision: binary64

Cost: 1604

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

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 8600000000000:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t_0 \cdot \left(t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

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 (+ 2.0 (+ alpha beta))))
   (if (<= beta 8600000000000.0)
     (/ (+ alpha (+ 1.0 beta)) (* t_0 (* t_0 (+ beta (+ alpha 3.0)))))
     (/
      (/ (+ 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 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 8600000000000.0) {
		tmp = (alpha + (1.0 + beta)) / (t_0 * (t_0 * (beta + (alpha + 3.0))));
	} 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 = 2.0d0 + (alpha + beta)
    if (beta <= 8600000000000.0d0) then
        tmp = (alpha + (1.0d0 + beta)) / (t_0 * (t_0 * (beta + (alpha + 3.0d0))))
    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 = 2.0 + (alpha + beta);
	double tmp;
	if (beta <= 8600000000000.0) {
		tmp = (alpha + (1.0 + beta)) / (t_0 * (t_0 * (beta + (alpha + 3.0))));
	} 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 = 2.0 + (alpha + beta)
	tmp = 0
	if beta <= 8600000000000.0:
		tmp = (alpha + (1.0 + beta)) / (t_0 * (t_0 * (beta + (alpha + 3.0))))
	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(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 8600000000000.0)
		tmp = Float64(Float64(alpha + Float64(1.0 + beta)) / Float64(t_0 * Float64(t_0 * Float64(beta + Float64(alpha + 3.0)))));
	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 = 2.0 + (alpha + beta);
	tmp = 0.0;
	if (beta <= 8600000000000.0)
		tmp = (alpha + (1.0 + beta)) / (t_0 * (t_0 * (beta + (alpha + 3.0))));
	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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8600000000000.0], N[(N[(alpha + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 8.6e12

   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%

      \[\leadsto \color{blue}{\frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}} \]
      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	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-/l/ [=>] 99.9	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 99.9	\[ \frac{\color{blue}{\left(\alpha + \beta\right) + \left(\beta \cdot \alpha + 1\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 99.9	\[ \frac{\color{blue}{\alpha + \left(\beta + \left(\beta \cdot \alpha + 1\right)\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{\alpha + \left(\beta + \left(\color{blue}{\alpha \cdot \beta} + 1\right)\right)}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      fma-def [=>] 99.9	\[ \frac{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(\alpha, \beta, 1\right)}\right)}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{\alpha + \left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right)}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   3. Taylor expanded in alpha around 0 99.7%

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

   if 8.6e12 < beta

   1. Initial program 90.2%

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

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8600000000000:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1600

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

Alternative 2
Accuracy	98.1%
Cost	1220

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

Alternative 3
Accuracy	98.8%
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(\alpha + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	964

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\frac{0.25 + \alpha \cdot \left(\alpha \cdot -0.0625\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	836

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	97.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	91.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	92.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	94.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{0.25}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 10
Accuracy	46.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2222222222222222}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	91.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{0.25}{\alpha + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Accuracy	46.3%
Cost	320

\[\frac{0.25}{\beta + 3} \]

Alternative 13
Accuracy	6.0%
Cost	192

\[\frac{0.2222222222222222}{\beta} \]

Reproduce

herbie shell --seed 2023139 
(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)))