Octave 3.8, jcobi/4

Average Error: 53.8 → 9.3

Time: 40.7s

Precision: binary64

Cost: 1604

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{1}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \left(\alpha + i\right)}}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+127)
   0.0625
   (*
    0.5
    (/
     (/ 1.0 (+ beta (+ alpha (+ i i))))
     (/ (+ 1.0 (/ beta i)) (* 2.0 (+ alpha i)))))))

C

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

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+127) {
		tmp = 0.0625;
	} else {
		tmp = 0.5 * ((1.0 / (beta + (alpha + (i + i)))) / ((1.0 + (beta / i)) / (2.0 * (alpha + i))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.8d+127) then
        tmp = 0.0625d0
    else
        tmp = 0.5d0 * ((1.0d0 / (beta + (alpha + (i + i)))) / ((1.0d0 + (beta / i)) / (2.0d0 * (alpha + i))))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+127) {
		tmp = 0.0625;
	} else {
		tmp = 0.5 * ((1.0 / (beta + (alpha + (i + i)))) / ((1.0 + (beta / i)) / (2.0 * (alpha + i))));
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.8e+127:
		tmp = 0.0625
	else:
		tmp = 0.5 * ((1.0 / (beta + (alpha + (i + i)))) / ((1.0 + (beta / i)) / (2.0 * (alpha + i))))
	return tmp

Julia

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

↓

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.8e+127)
		tmp = 0.0625;
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 / Float64(beta + Float64(alpha + Float64(i + i)))) / Float64(Float64(1.0 + Float64(beta / i)) / Float64(2.0 * Float64(alpha + i)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.8e+127)
		tmp = 0.0625;
	else
		tmp = 0.5 * ((1.0 / (beta + (alpha + (i + i)))) / ((1.0 + (beta / i)) / (2.0 * (alpha + i))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.8e+127], 0.0625, N[(0.5 * N[(N[(1.0 / N[(beta + N[(alpha + N[(i + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(beta / i), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998e127

   1. Initial program 48.2

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

   2. Simplified 31.9

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

      [Start] 48.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-50 [=>] 48.2	\[ \color{blue}{\frac{-\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{1 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]
      rational.json-simplify-10 [=>] 48.2	\[ \frac{\color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{-1}}}{1 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \]
      rational.json-simplify-47 [=>] 48.2	\[ \color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{-1 \cdot \left(1 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

   3. Taylor expanded in i around inf 4.7

      \[\leadsto \color{blue}{0.0625} \]

   if 3.7999999999999998e127 < beta

   1. Initial program 63.7

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

   2. Simplified 54.6

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

      [Start] 63.7	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-49 [=>] 54.6	\[ \frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-49 [=>] 54.6	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 46.5

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

   4. Applied egg-rr 33.2

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

   5. Simplified 33.2

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

      [Start] 33.2	\[ \frac{i}{\frac{\beta}{i + \alpha} \cdot \left(\left(\beta + \left(i + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta + \left(i + \left(i + \alpha\right)\right)}{i + \left(\alpha + \beta\right)}\right)} \]
      rational.json-simplify-43 [=>] 33.2	\[ \frac{i}{\color{blue}{\left(\beta + \left(i + \left(i + \alpha\right)\right)\right) \cdot \left(\frac{\beta + \left(i + \left(i + \alpha\right)\right)}{i + \left(\alpha + \beta\right)} \cdot \frac{\beta}{i + \alpha}\right)}} \]
      rational.json-simplify-43 [=>] 33.2	\[ \frac{i}{\color{blue}{\frac{\beta + \left(i + \left(i + \alpha\right)\right)}{i + \left(\alpha + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)}} \]
      rational.json-simplify-41 [=>] 33.2	\[ \frac{i}{\frac{\color{blue}{i + \left(\left(i + \alpha\right) + \beta\right)}}{i + \left(\alpha + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 33.2	\[ \frac{i}{\frac{i + \color{blue}{\left(\beta + \left(i + \alpha\right)\right)}}{i + \left(\alpha + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-41 [<=] 33.2	\[ \frac{i}{\frac{i + \color{blue}{\left(\alpha + \left(\beta + i\right)\right)}}{i + \left(\alpha + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \color{blue}{\left(i + \beta\right)}\right)}{i + \left(\alpha + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-41 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\color{blue}{\alpha + \left(\beta + i\right)}} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \color{blue}{\left(i + \beta\right)}} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(\beta + \left(i + \left(i + \alpha\right)\right)\right)\right)} \]
      rational.json-simplify-41 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(i + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \color{blue}{\left(i + \left(\left(i + \alpha\right) + \beta\right)\right)}\right)} \]
      rational.json-simplify-1 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(i + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(i + \color{blue}{\left(\beta + \left(i + \alpha\right)\right)}\right)\right)} \]
      rational.json-simplify-41 [<=] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(i + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(i + \color{blue}{\left(\alpha + \left(\beta + i\right)\right)}\right)\right)} \]
      rational.json-simplify-1 [=>] 33.2	\[ \frac{i}{\frac{i + \left(\alpha + \left(i + \beta\right)\right)}{\alpha + \left(i + \beta\right)} \cdot \left(\frac{\beta}{i + \alpha} \cdot \left(i + \left(\alpha + \color{blue}{\left(i + \beta\right)}\right)\right)\right)} \]

   6. Taylor expanded in beta around inf 31.6

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

   7. Applied egg-rr 33.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \left(i + \alpha\right)}{\frac{\beta + \left(i + \left(i + \alpha\right)\right)}{\frac{1}{1 + \frac{\beta}{i}}}}} \]

   8. Simplified 17.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{1}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \left(\alpha + i\right)}}} \]
      Proof

      [Start] 33.0	\[ 0.5 \cdot \frac{2 \cdot \left(i + \alpha\right)}{\frac{\beta + \left(i + \left(i + \alpha\right)\right)}{\frac{1}{1 + \frac{\beta}{i}}}} \]
      rational.json-simplify-61 [=>] 33.0	\[ 0.5 \cdot \frac{2 \cdot \left(i + \alpha\right)}{\color{blue}{\frac{1 + \frac{\beta}{i}}{\frac{1}{\beta + \left(i + \left(i + \alpha\right)\right)}}}} \]
      rational.json-simplify-61 [=>] 17.6	\[ 0.5 \cdot \color{blue}{\frac{\frac{1}{\beta + \left(i + \left(i + \alpha\right)\right)}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \left(i + \alpha\right)}}} \]
      rational.json-simplify-41 [<=] 17.6	\[ 0.5 \cdot \frac{\frac{1}{\beta + \color{blue}{\left(\alpha + \left(i + i\right)\right)}}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \left(i + \alpha\right)}} \]
      rational.json-simplify-1 [=>] 17.6	\[ 0.5 \cdot \frac{\frac{1}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \color{blue}{\left(\alpha + i\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{1}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\frac{1 + \frac{\beta}{i}}{2 \cdot \left(\alpha + i\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.3
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+127}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{1 + \frac{\beta}{i}} \cdot \frac{1}{\beta + \left(i + \left(i + \alpha\right)\right)}\\ \end{array} \]

Alternative 2
Error	9.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.58 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta + \alpha} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+148}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Error	18.9
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023075 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))