Octave 3.8, jcobi/3

Average Error: 5.59% → 0.72%

Time: 23.3s

Precision: binary64

Cost: 1736

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\alpha \leq -2.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_1 \cdot \left(\alpha + 2\right)}}{t_0}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t_1 \cdot \left(1 - \frac{-1 - \alpha}{\beta}\right)}}{t_0}\\ \end{array} \]

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 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= alpha -2.9e-14)
     (/ (/ (+ alpha 1.0) (* t_1 (+ alpha 2.0))) t_0)
     (if (<= alpha 3.9e-23)
       (* (/ (+ beta 1.0) (+ beta 3.0)) (/ (/ 1.0 (+ beta 2.0)) (+ beta 2.0)))
       (/ (/ (+ alpha 1.0) (* t_1 (- 1.0 (/ (- -1.0 alpha) beta)))) t_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 + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (alpha <= -2.9e-14) {
		tmp = ((alpha + 1.0) / (t_1 * (alpha + 2.0))) / t_0;
	} else if (alpha <= 3.9e-23) {
		tmp = ((beta + 1.0) / (beta + 3.0)) * ((1.0 / (beta + 2.0)) / (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (t_1 * (1.0 - ((-1.0 - alpha) / beta)))) / t_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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (alpha <= (-2.9d-14)) then
        tmp = ((alpha + 1.0d0) / (t_1 * (alpha + 2.0d0))) / t_0
    else if (alpha <= 3.9d-23) then
        tmp = ((beta + 1.0d0) / (beta + 3.0d0)) * ((1.0d0 / (beta + 2.0d0)) / (beta + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / (t_1 * (1.0d0 - (((-1.0d0) - alpha) / beta)))) / t_0
    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 + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (alpha <= -2.9e-14) {
		tmp = ((alpha + 1.0) / (t_1 * (alpha + 2.0))) / t_0;
	} else if (alpha <= 3.9e-23) {
		tmp = ((beta + 1.0) / (beta + 3.0)) * ((1.0 / (beta + 2.0)) / (beta + 2.0));
	} else {
		tmp = ((alpha + 1.0) / (t_1 * (1.0 - ((-1.0 - alpha) / beta)))) / t_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 + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if alpha <= -2.9e-14:
		tmp = ((alpha + 1.0) / (t_1 * (alpha + 2.0))) / t_0
	elif alpha <= 3.9e-23:
		tmp = ((beta + 1.0) / (beta + 3.0)) * ((1.0 / (beta + 2.0)) / (beta + 2.0))
	else:
		tmp = ((alpha + 1.0) / (t_1 * (1.0 - ((-1.0 - alpha) / beta)))) / t_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 + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (alpha <= -2.9e-14)
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(t_1 * Float64(alpha + 2.0))) / t_0);
	elseif (alpha <= 3.9e-23)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(beta + 3.0)) * Float64(Float64(1.0 / Float64(beta + 2.0)) / Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(t_1 * Float64(1.0 - Float64(Float64(-1.0 - alpha) / beta)))) / t_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 + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (alpha <= -2.9e-14)
		tmp = ((alpha + 1.0) / (t_1 * (alpha + 2.0))) / t_0;
	elseif (alpha <= 3.9e-23)
		tmp = ((beta + 1.0) / (beta + 3.0)) * ((1.0 / (beta + 2.0)) / (beta + 2.0));
	else
		tmp = ((alpha + 1.0) / (t_1 * (1.0 - ((-1.0 - alpha) / beta)))) / t_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 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -2.9e-14], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$1 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[alpha, 3.9e-23], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$1 * N[(1.0 - N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.9000000000000003e-14

   1. Initial program 0.71

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

   2. Simplified 0.55

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

      [Start] 0.71	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Taylor expanded in beta around 0 16.58

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

   if -2.9000000000000003e-14 < alpha < 3.9e-23

   1. Initial program 0.15

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

   2. Simplified 0.57

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

      [Start] 0.15	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 0.57	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 0.57	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta\right) + \left(\beta \cdot \alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\color{blue}{\alpha \cdot \beta} + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      metadata-eval [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]
      associate-+l+ [=>] 0.57	\[ \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

   3. Taylor expanded in alpha around 0 0.59

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

   4. Taylor expanded in alpha around 0 0.59

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

   5. Applied egg-rr 0.17

      \[\leadsto \color{blue}{\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}} \]

   if 3.9e-23 < alpha

   1. Initial program 42.55

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

   2. Simplified 0.3

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

      [Start] 42.55	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Applied egg-rr 0.32

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

   4. Taylor expanded in beta around -inf 0.64

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

   5. Simplified 0.64

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

      [Start] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + -1 \cdot \frac{1 + -1 \cdot \left(2 + \alpha\right)}{\beta}\right)}}{\alpha + \left(\beta + 3\right)} \]
      mul-1-neg [=>] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(2 + \alpha\right)}{\beta}\right)}\right)}}{\alpha + \left(\beta + 3\right)} \]
      distribute-lft-in [=>] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{1 + \color{blue}{\left(-1 \cdot 2 + -1 \cdot \alpha\right)}}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{1 + \left(\color{blue}{-2} + -1 \cdot \alpha\right)}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      neg-mul-1 [<=] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{1 + \left(-2 + \color{blue}{\left(-\alpha\right)}\right)}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      sub-neg [<=] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{1 + \color{blue}{\left(-2 - \alpha\right)}}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      associate-+r- [=>] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{\color{blue}{\left(1 + -2\right) - \alpha}}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]
      metadata-eval [=>] 0.64	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 + \left(-\frac{\color{blue}{-1} - \alpha}{\beta}\right)\right)}}{\alpha + \left(\beta + 3\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.72

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 - \frac{-1 - \alpha}{\beta}\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.18%
Cost	1600

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

Alternative 2
Error	1.04%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -2.45 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3
Error	1.03%
Cost	1352

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\alpha \leq -3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + 2\right)}}{t_0}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 4
Error	1.7%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(\alpha + 2\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 5
Error	1.43%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 3} \cdot \frac{\frac{1}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Error	1.72%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	1.69%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(\alpha + 2\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 8
Error	1.69%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{-1 - \beta}{\beta + 3}}{-2 - \beta}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(\alpha + 2\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 9
Error	2.92%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	6.26%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11
Error	3.53%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12
Error	2.97%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	53.26%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 14
Error	8.68%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	8.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	53.69%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 17
Error	55.42%
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023102 
(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)))