Octave 3.8, jcobi/2

Average Error: 24.0 → 1.3

Time: 1.5min

Precision: binary64

Cost: 4036

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + \left(i + i\right)\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_1 + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{\beta}{t_0} - \alpha \cdot \frac{\alpha}{t_0}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)} \cdot 0.5 - -0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha (+ i i)))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) -0.999)
     (* 0.5 (/ (+ (* i 4.0) (+ 2.0 (* 2.0 beta))) alpha))
     (-
      (*
       (/
        (- (* beta (/ beta t_0)) (* alpha (/ alpha t_0)))
        (+ alpha (+ (+ i i) (+ beta 2.0))))
       0.5)
      -0.5))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = beta + (alpha + (i + i));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) <= -0.999) {
		tmp = 0.5 * (((i * 4.0) + (2.0 + (2.0 * beta))) / alpha);
	} else {
		tmp = ((((beta * (beta / t_0)) - (alpha * (alpha / t_0))) / (alpha + ((i + i) + (beta + 2.0)))) * 0.5) - -0.5;
	}
	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 = (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0d0 * i))) / (((alpha + beta) + (2.0d0 * i)) + 2.0d0)) + 1.0d0) / 2.0d0
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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + (i + i))
    t_1 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0d0)) <= (-0.999d0)) then
        tmp = 0.5d0 * (((i * 4.0d0) + (2.0d0 + (2.0d0 * beta))) / alpha)
    else
        tmp = ((((beta * (beta / t_0)) - (alpha * (alpha / t_0))) / (alpha + ((i + i) + (beta + 2.0d0)))) * 0.5d0) - (-0.5d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (alpha + (i + i));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) <= -0.999) {
		tmp = 0.5 * (((i * 4.0) + (2.0 + (2.0 * beta))) / alpha);
	} else {
		tmp = ((((beta * (beta / t_0)) - (alpha * (alpha / t_0))) / (alpha + ((i + i) + (beta + 2.0)))) * 0.5) - -0.5;
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta, i):
	t_0 = beta + (alpha + (i + i))
	t_1 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) <= -0.999:
		tmp = 0.5 * (((i * 4.0) + (2.0 + (2.0 * beta))) / alpha)
	else:
		tmp = ((((beta * (beta / t_0)) - (alpha * (alpha / t_0))) / (alpha + ((i + i) + (beta + 2.0)))) * 0.5) - -0.5
	return tmp

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(alpha + Float64(i + i)))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) <= -0.999)
		tmp = Float64(0.5 * Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(2.0 * beta))) / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta * Float64(beta / t_0)) - Float64(alpha * Float64(alpha / t_0))) / Float64(alpha + Float64(Float64(i + i) + Float64(beta + 2.0)))) * 0.5) - -0.5);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (alpha + (i + i));
	t_1 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) <= -0.999)
		tmp = 0.5 * (((i * 4.0) + (2.0 + (2.0 * beta))) / alpha);
	else
		tmp = ((((beta * (beta / t_0)) - (alpha * (alpha / t_0))) / (alpha + ((i + i) + (beta + 2.0)))) * 0.5) - -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(alpha + N[(i + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(0.5 * N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(beta * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha * N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(N[(i + i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.998999999999999999

   1. Initial program 61.9

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

   2. Simplified 62.0

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

      [Start] 61.9	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational_best-simplify-65 [=>] 61.9	\[ \color{blue}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{2} + \frac{1}{2}} \]

   3. Taylor expanded in alpha around inf 5.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}} \]

   4. Simplified 5.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}} \]
      Proof

      [Start] 5.7	\[ 0.5 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} \]
      rational_best-simplify-1 [=>] 5.7	\[ 0.5 \cdot \frac{\color{blue}{i \cdot 4} + \left(2 + 2 \cdot \beta\right)}{\alpha} \]

   if -0.998999999999999999 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 12.7

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

   2. Simplified 12.7

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

      [Start] 12.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational_best-simplify-65 [=>] 12.7	\[ \color{blue}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{2} + \frac{1}{2}} \]
      rational_best-simplify-53 [=>] 12.7	\[ \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot 2}} + \frac{1}{2} \]
      rational_best-simplify-110 [=>] 12.7	\[ \frac{\frac{\color{blue}{\beta \cdot \beta - \alpha \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-3 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-3 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-47 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-3 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\color{blue}{\left(2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-3 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\left(2 + \color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)}\right) \cdot 2} + \frac{1}{2} \]
      rational_best-simplify-47 [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot 2} + \frac{1}{2} \]
      metadata-eval [=>] 12.7	\[ \frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + \color{blue}{0.5} \]

   3. Applied egg-rr 1.1

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

   4. Simplified 1.1

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

      [Start] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\beta + \left(\alpha + \left(i + i\right)\right)} - \frac{\alpha \cdot \alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-47 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\color{blue}{\left(i + i\right) + \left(\alpha + \beta\right)}} - \frac{\alpha \cdot \alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [<=] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\left(i + i\right) + \color{blue}{\left(\beta + \alpha\right)}} - \frac{\alpha \cdot \alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\color{blue}{\left(\beta + \alpha\right) + \left(i + i\right)}} - \frac{\alpha \cdot \alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\color{blue}{\left(\alpha + \beta\right)} + \left(i + i\right)} - \frac{\alpha \cdot \alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-47 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\left(\alpha + \beta\right) + \left(i + i\right)} - \frac{\alpha \cdot \alpha}{\color{blue}{\left(i + i\right) + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [<=] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\left(\alpha + \beta\right) + \left(i + i\right)} - \frac{\alpha \cdot \alpha}{\left(i + i\right) + \color{blue}{\left(\beta + \alpha\right)}}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\left(\alpha + \beta\right) + \left(i + i\right)} - \frac{\alpha \cdot \alpha}{\color{blue}{\left(\beta + \alpha\right) + \left(i + i\right)}}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]
      rational_best-simplify-3 [=>] 1.1	\[ \frac{\beta \cdot \frac{\beta}{\left(\alpha + \beta\right) + \left(i + i\right)} - \frac{\alpha \cdot \alpha}{\color{blue}{\left(\alpha + \beta\right)} + \left(i + i\right)}}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5 \]

   5. Applied egg-rr 0.0

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

   6. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{\beta \cdot \frac{\beta}{\beta + \left(\alpha + \left(i + i\right)\right)} - \alpha \cdot \frac{\alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)} \cdot 0.5 - -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{\beta}{\beta + \left(\alpha + \left(i + i\right)\right)} - \alpha \cdot \frac{\alpha}{\beta + \left(\alpha + \left(i + i\right)\right)}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)} \cdot 0.5 - -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	4036

\[\begin{array}{l} t_0 := i + \left(i + \left(\alpha + \beta\right)\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_1 + 2} \leq -0.999:\\ \;\;\;\;0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\beta \cdot \frac{\beta}{t_0} - \alpha \cdot \frac{\alpha}{t_0}}{\left(\alpha + \beta\right) + \left(2 + \left(i + i\right)\right)} - -0.5\\ \end{array} \]

Alternative 2
Error	1.6
Cost	3396

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.5:\\ \;\;\;\;0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{\beta}{\left(\alpha + \beta\right) + \left(i + i\right)} - \alpha}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5\\ \end{array} \]

Alternative 3
Error	12.7
Cost	1624

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ t_1 := 0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ t_2 := -0.5 \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)} + 0.5\\ \mathbf{if}\;\alpha \leq -9.8 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 13:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + 0.5\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.7
Cost	1368

\[\begin{array}{l} t_0 := \left(2 + i \cdot 4\right) \cdot \frac{0.5}{\alpha}\\ t_1 := 0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ t_2 := -0.5 \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)} + 0.5\\ \mathbf{if}\;\alpha \leq -8 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 2.2 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 0.044:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 2.15 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.6
Cost	1368

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ t_1 := \left(2 + i \cdot 4\right) \cdot \frac{0.5}{\alpha}\\ t_2 := -0.5 \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)} + 0.5\\ \mathbf{if}\;\alpha \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + 0.5\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.1
Cost	1360

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ t_1 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\alpha \leq 14.6:\\ \;\;\;\;0.5 \cdot \frac{\beta - \alpha}{t_1} + 0.5\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \frac{\beta}{t_1} + 0.5\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	7.5
Cost	1228

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{i \cdot 4 + \left(2 + 2 \cdot \beta\right)}{\alpha}\\ \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\beta}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot 2} + 0.5\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	13.2
Cost	972

\[\begin{array}{l} t_0 := \left(2 + i \cdot 4\right) \cdot \frac{0.5}{\alpha}\\ t_1 := 0.5 \cdot \frac{\beta}{\beta + 2} + 0.5\\ \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3.25 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	18.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+96}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	24.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))