Octave 3.8, jcobi/3

Average Accuracy: 94.5% → 99.5%

Time: 16.7s

Precision: binary64

Cost: 1732

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(\alpha + 1\right)}{t_0 \cdot \left(t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

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 5e+88)
     (/ (* (+ 1.0 beta) (+ alpha 1.0)) (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) beta) (+ beta (+ alpha 2.0))))))

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 <= 5e+88) {
		tmp = ((1.0 + beta) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 2.0));
	}
	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 <= 5d+88) then
        tmp = ((1.0d0 + beta) * (alpha + 1.0d0)) / (t_0 * (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((alpha + 1.0d0) / beta) / (beta + (alpha + 2.0d0))
    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 <= 5e+88) {
		tmp = ((1.0 + beta) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 2.0));
	}
	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 <= 5e+88:
		tmp = ((1.0 + beta) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 2.0))
	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 <= 5e+88)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(alpha + 1.0)) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + Float64(alpha + 2.0)));
	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 <= 5e+88)
		tmp = ((1.0 + beta) * (alpha + 1.0)) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = ((alpha + 1.0) / beta) / (beta + (alpha + 2.0));
	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, 5e+88], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999997e88

   1. Initial program 99.3%

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

   2. Simplified 92.6%

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

   if 4.99999999999999997e88 < beta

   1. Initial program 81.2%

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

   2. Simplified 91.5%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in beta around inf 94.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\left(1 + \beta\right) \cdot \frac{\frac{\alpha + 1}{t_0}}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	1600

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

Alternative 3
Accuracy	97.6%
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{t_0}\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	1220

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.05:\\ \;\;\;\;\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(\alpha + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{1 + t_0}\\ \end{array} \]

Alternative 5
Accuracy	97.2%
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_0}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	97.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	97.2%
Cost	836

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	96.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	96.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	91.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	45.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	90.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.25:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13
Accuracy	91.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	45.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \]

Alternative 15
Accuracy	44.2%
Cost	320

\[\frac{0.16666666666666666}{\alpha + 2} \]

Alternative 16
Accuracy	44.0%
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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)))