Octave 3.8, jcobi/2

Average Error: 24.6 → 1.5

Time: 31.0s

Precision: binary64

Cost: 17988

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 + 2 \cdot i\\ t_1 := \beta + t_0\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := \left(-\left(\beta + 2 \cdot i\right)\right) - t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{t_2 + 2} \leq -0.5:\\ \;\;\;\;\frac{i \cdot \left(\frac{\beta}{{\alpha}^{2}} \cdot -2\right) + \left(\left(-\frac{t_3}{\alpha}\right) - \left(-t_1 \cdot \frac{t_3}{{\alpha}^{2}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + t_0} + 1}{2}\\ \end{array} \]

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 (+ 2.0 (* 2.0 i)))
        (t_1 (+ beta t_0))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (- (- (+ beta (* 2.0 i))) t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ t_2 2.0)) -0.5)
     (/
      (+
       (* i (* (/ beta (pow alpha 2.0)) -2.0))
       (- (- (/ t_3 alpha)) (- (* t_1 (/ t_3 (pow alpha 2.0))))))
      2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
        (/ (- beta alpha) (+ (+ alpha beta) t_0)))
       1.0)
      2.0))))

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 = 2.0 + (2.0 * i);
	double t_1 = beta + t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = -(beta + (2.0 * i)) - t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5) {
		tmp = ((i * ((beta / pow(alpha, 2.0)) * -2.0)) + (-(t_3 / alpha) - -(t_1 * (t_3 / pow(alpha, 2.0))))) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + t_0))) + 1.0) / 2.0;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 2.0d0 + (2.0d0 * i)
    t_1 = beta + t_0
    t_2 = (alpha + beta) + (2.0d0 * i)
    t_3 = -(beta + (2.0d0 * i)) - t_1
    if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0d0)) <= (-0.5d0)) then
        tmp = ((i * ((beta / (alpha ** 2.0d0)) * (-2.0d0))) + (-(t_3 / alpha) - -(t_1 * (t_3 / (alpha ** 2.0d0))))) / 2.0d0
    else
        tmp = ((((alpha + beta) / (beta + (alpha + (2.0d0 * i)))) * ((beta - alpha) / ((alpha + beta) + t_0))) + 1.0d0) / 2.0d0
    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 = 2.0 + (2.0 * i);
	double t_1 = beta + t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = -(beta + (2.0 * i)) - t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5) {
		tmp = ((i * ((beta / Math.pow(alpha, 2.0)) * -2.0)) + (-(t_3 / alpha) - -(t_1 * (t_3 / Math.pow(alpha, 2.0))))) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + t_0))) + 1.0) / 2.0;
	}
	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 = 2.0 + (2.0 * i)
	t_1 = beta + t_0
	t_2 = (alpha + beta) + (2.0 * i)
	t_3 = -(beta + (2.0 * i)) - t_1
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5:
		tmp = ((i * ((beta / math.pow(alpha, 2.0)) * -2.0)) + (-(t_3 / alpha) - -(t_1 * (t_3 / math.pow(alpha, 2.0))))) / 2.0
	else:
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + t_0))) + 1.0) / 2.0
	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(2.0 + Float64(2.0 * i))
	t_1 = Float64(beta + t_0)
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = Float64(Float64(-Float64(beta + Float64(2.0 * i))) - t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(t_2 + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(i * Float64(Float64(beta / (alpha ^ 2.0)) * -2.0)) + Float64(Float64(-Float64(t_3 / alpha)) - Float64(-Float64(t_1 * Float64(t_3 / (alpha ^ 2.0)))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(beta + Float64(alpha + Float64(2.0 * i)))) * Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + t_0))) + 1.0) / 2.0);
	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 = 2.0 + (2.0 * i);
	t_1 = beta + t_0;
	t_2 = (alpha + beta) + (2.0 * i);
	t_3 = -(beta + (2.0 * i)) - t_1;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5)
		tmp = ((i * ((beta / (alpha ^ 2.0)) * -2.0)) + (-(t_3 / alpha) - -(t_1 * (t_3 / (alpha ^ 2.0))))) / 2.0;
	else
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + t_0))) + 1.0) / 2.0;
	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[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]) - t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(i * N[(N[(beta / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[((-N[(t$95$3 / alpha), $MachinePrecision]) - (-N[(t$95$1 * N[(t$95$3 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $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)) < -0.5

   1. Initial program 61.3

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

   2. Simplified 53.2

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

      [Start] 61.3	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 53.2	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 53.2	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      rational.json-simplify-1 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      rational.json-simplify-41 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} + 1}{2} \]
      rational.json-simplify-1 [=>] 53.2	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 2 \cdot i\right)}} + 1}{2} \]

   3. Taylor expanded in alpha around inf 57.7

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

   4. Taylor expanded in alpha around -inf 13.6

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

   5. Simplified 6.2

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

      [Start] 13.6	\[ \frac{\left(-2 \cdot \frac{\beta \cdot i}{{\alpha}^{2}} + -1 \cdot \frac{-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha}\right) - -1 \cdot \frac{\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}}{2} \]
      rational.json-simplify-1 [=>] 13.6	\[ \frac{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -2 \cdot \frac{\beta \cdot i}{{\alpha}^{2}}\right)} - -1 \cdot \frac{\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}}{2} \]
      rational.json-simplify-48 [=>] 13.6	\[ \frac{\color{blue}{-2 \cdot \frac{\beta \cdot i}{{\alpha}^{2}} + \left(-1 \cdot \frac{-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} - -1 \cdot \frac{\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)}}{2} \]
      rational.json-simplify-49 [=>] 13.6	\[ \frac{-2 \cdot \color{blue}{\left(i \cdot \frac{\beta}{{\alpha}^{2}}\right)} + \left(-1 \cdot \frac{-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} - -1 \cdot \frac{\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)}{2} \]
      rational.json-simplify-43 [=>] 13.6	\[ \frac{\color{blue}{i \cdot \left(\frac{\beta}{{\alpha}^{2}} \cdot -2\right)} + \left(-1 \cdot \frac{-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} - -1 \cdot \frac{\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)}{2} \]

   if -0.5 < (/.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 13.4

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

   2. Simplified 0.0

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

      [Start] 13.4	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 0.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      rational.json-simplify-41 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 2 \cdot i\right)}} + 1}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.4
Cost	3524

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternative 2
Error	7.4
Cost	1868

\[\begin{array}{l} t_0 := \frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{2 \cdot i + \beta}}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ t_1 := \frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 3 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3 \cdot 10^{+162}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.9
Cost	1236

\[\begin{array}{l} t_0 := \frac{\frac{2}{2 + \alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;i \leq 8.2 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.25 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4
Error	11.3
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.8
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 4.7 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	23.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.8 \cdot 10^{+85}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+255}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 7
Error	16.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	17.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+64}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	24.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023075 
(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))