Octave 3.8, jcobi/2

Average Error: 24.2 → 1.5

Time: 31.7s

Precision: binary64

Cost: 18820

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

Math

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

↓

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

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 (+ beta (* i 2.0))))
        (t_1 (+ (* i -2.0) (- t_0)))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ t_2 2.0)) -0.5)
     (/
      (+
       (* t_1 (/ (- beta t_1) (pow alpha 2.0)))
       (-
        (+ (* 2.0 (* i (/ t_0 (pow alpha 2.0)))) (/ beta alpha))
        (+ (- (/ t_0 alpha)) (* (/ i alpha) -2.0))))
      2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
        (/ (- beta alpha) (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))))
       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 + (beta + (i * 2.0));
	double t_1 = (i * -2.0) + -t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5) {
		tmp = ((t_1 * ((beta - t_1) / pow(alpha, 2.0))) + (((2.0 * (i * (t_0 / pow(alpha, 2.0)))) + (beta / alpha)) - (-(t_0 / alpha) + ((i / alpha) * -2.0)))) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 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) :: tmp
    t_0 = 2.0d0 + (beta + (i * 2.0d0))
    t_1 = (i * (-2.0d0)) + -t_0
    t_2 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0d0)) <= (-0.5d0)) then
        tmp = ((t_1 * ((beta - t_1) / (alpha ** 2.0d0))) + (((2.0d0 * (i * (t_0 / (alpha ** 2.0d0)))) + (beta / alpha)) - (-(t_0 / alpha) + ((i / alpha) * (-2.0d0))))) / 2.0d0
    else
        tmp = ((((alpha + beta) / (beta + (alpha + (2.0d0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) + 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 + (beta + (i * 2.0));
	double t_1 = (i * -2.0) + -t_0;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5) {
		tmp = ((t_1 * ((beta - t_1) / Math.pow(alpha, 2.0))) + (((2.0 * (i * (t_0 / Math.pow(alpha, 2.0)))) + (beta / alpha)) - (-(t_0 / alpha) + ((i / alpha) * -2.0)))) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 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 + (beta + (i * 2.0))
	t_1 = (i * -2.0) + -t_0
	t_2 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5:
		tmp = ((t_1 * ((beta - t_1) / math.pow(alpha, 2.0))) + (((2.0 * (i * (t_0 / math.pow(alpha, 2.0)))) + (beta / alpha)) - (-(t_0 / alpha) + ((i / alpha) * -2.0)))) / 2.0
	else:
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 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(beta + Float64(i * 2.0)))
	t_1 = Float64(Float64(i * -2.0) + Float64(-t_0))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	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(t_1 * Float64(Float64(beta - t_1) / (alpha ^ 2.0))) + Float64(Float64(Float64(2.0 * Float64(i * Float64(t_0 / (alpha ^ 2.0)))) + Float64(beta / alpha)) - Float64(Float64(-Float64(t_0 / alpha)) + Float64(Float64(i / 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) + Float64(2.0 + Float64(2.0 * i))))) + 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 + (beta + (i * 2.0));
	t_1 = (i * -2.0) + -t_0;
	t_2 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) <= -0.5)
		tmp = ((t_1 * ((beta - t_1) / (alpha ^ 2.0))) + (((2.0 * (i * (t_0 / (alpha ^ 2.0)))) + (beta / alpha)) - (-(t_0 / alpha) + ((i / alpha) * -2.0)))) / 2.0;
	else
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * -2.0), $MachinePrecision] + (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = 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$2), $MachinePrecision] / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(t$95$1 * N[(N[(beta - t$95$1), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(i * N[(t$95$0 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[((-N[(t$95$0 / alpha), $MachinePrecision]) + N[(N[(i / alpha), $MachinePrecision] * -2.0), $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] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $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 52.9

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}} \]
      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 [=>] 52.9	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 52.9	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      rational.json-simplify-1 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      rational.json-simplify-41 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} + 1}{2} \]
      rational.json-simplify-1 [=>] 52.9	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 2 \cdot i\right)}} + 1}{2} \]

   3. Taylor expanded in beta around 0 52.9

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

   4. Simplified 52.9

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

      [Start] 52.9	\[ \frac{\frac{\alpha}{\alpha + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2} \]
      rational.json-simplify-1 [=>] 52.9	\[ \frac{\frac{\alpha}{\color{blue}{2 \cdot i + \alpha}} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2} \]

   5. Taylor expanded in alpha around -inf 13.4

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

   6. Simplified 6.2

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

      [Start] 13.4	\[ \frac{\left(\frac{\left(\beta - \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right) \cdot \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}} + \left(\frac{\beta}{\alpha} + 2 \cdot \frac{i \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha} + -2 \cdot \frac{i}{\alpha}\right)}{2} \]
      rational.json-simplify-1 [=>] 13.4	\[ \frac{\color{blue}{\left(\left(\frac{\beta}{\alpha} + 2 \cdot \frac{i \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right) + \frac{\left(\beta - \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right) \cdot \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right)} - \left(-1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha} + -2 \cdot \frac{i}{\alpha}\right)}{2} \]
      rational.json-simplify-48 [=>] 13.4	\[ \frac{\color{blue}{\frac{\left(\beta - \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right) \cdot \left(-2 \cdot i + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}} + \left(\left(\frac{\beta}{\alpha} + 2 \cdot \frac{i \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{\beta + \left(2 + 2 \cdot i\right)}{\alpha} + -2 \cdot \frac{i}{\alpha}\right)\right)}}{2} \]

   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 12.9

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

   2. Simplified 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] 12.9	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.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

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

Alternatives

Alternative 1
Error	1.4
Cost	3524

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.99999996:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternative 2
Error	7.6
Cost	1604

\[\begin{array}{l} t_0 := 2 + 2 \cdot \beta\\ \mathbf{if}\;\alpha \leq 1.38 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{2 \cdot i + \beta}}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{t_0}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + t_0}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	7.9
Cost	1356

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ t_1 := 2 + 2 \cdot \beta\\ \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{t_1}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 9.6 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + t_1}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.9
Cost	1356

\[\begin{array}{l} t_0 := 2 + 2 \cdot \beta\\ \mathbf{if}\;\alpha \leq 3.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{2 \cdot i + \beta}}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.15 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{t_0}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + t_0}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	11.3
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.5
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 5.3 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \frac{i}{\frac{\alpha}{4}}}{2}\\ \end{array} \]

Alternative 7
Error	17.2
Cost	980

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 5.7 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.6 \cdot 10^{+156}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 8.8 \cdot 10^{+213}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{+238}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.8
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+170}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	13.5
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	17.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+45}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	25.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023067 
(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))