Octave 3.8, jcobi/2

Average Error: 24.5 → 1.3

Time: 32.1s

Precision: binary64

Cost: 4548

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\left(-\left(i \cdot 4 + \left(2 + 2 \cdot \beta\right)\right)\right) - \left(\beta + \left(-\beta\right)\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

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 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) -0.98)
     (/
      (*
       (/ (+ beta alpha) (+ beta (+ (+ i i) alpha)))
       (*
        (/ (- beta alpha) (+ alpha (+ (+ i i) (+ 2.0 beta))))
        (/
         (- (- (+ (* i 4.0) (+ 2.0 (* 2.0 beta)))) (+ beta (- beta)))
         alpha)))
      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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.98) {
		tmp = (((beta + alpha) / (beta + ((i + i) + alpha))) * (((beta - alpha) / (alpha + ((i + i) + (2.0 + beta)))) * ((-((i * 4.0) + (2.0 + (2.0 * beta))) - (beta + -beta)) / alpha))) / 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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) <= (-0.98d0)) then
        tmp = (((beta + alpha) / (beta + ((i + i) + alpha))) * (((beta - alpha) / (alpha + ((i + i) + (2.0d0 + beta)))) * ((-((i * 4.0d0) + (2.0d0 + (2.0d0 * beta))) - (beta + -beta)) / alpha))) / 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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.98) {
		tmp = (((beta + alpha) / (beta + ((i + i) + alpha))) * (((beta - alpha) / (alpha + ((i + i) + (2.0 + beta)))) * ((-((i * 4.0) + (2.0 + (2.0 * beta))) - (beta + -beta)) / alpha))) / 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 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.98:
		tmp = (((beta + alpha) / (beta + ((i + i) + alpha))) * (((beta - alpha) / (alpha + ((i + i) + (2.0 + beta)))) * ((-((i * 4.0) + (2.0 + (2.0 * beta))) - (beta + -beta)) / alpha))) / 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(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) <= -0.98)
		tmp = Float64(Float64(Float64(Float64(beta + alpha) / Float64(beta + Float64(Float64(i + i) + alpha))) * Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(Float64(i + i) + Float64(2.0 + beta)))) * Float64(Float64(Float64(-Float64(Float64(i * 4.0) + Float64(2.0 + Float64(2.0 * beta)))) - Float64(beta + Float64(-beta))) / alpha))) / 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 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.98)
		tmp = (((beta + alpha) / (beta + ((i + i) + alpha))) * (((beta - alpha) / (alpha + ((i + i) + (2.0 + beta)))) * ((-((i * 4.0) + (2.0 + (2.0 * beta))) - (beta + -beta)) / alpha))) / 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[(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$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(N[(beta + alpha), $MachinePrecision] / N[(beta + N[(N[(i + i), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(N[(i + i), $MachinePrecision] + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[((-N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - N[(beta + (-beta)), $MachinePrecision]), $MachinePrecision] / alpha), $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)) < -0.97999999999999998

   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 54.3

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

      [Start] 61.9	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 54.3	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 54.3	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      rational.json-simplify-1 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      rational.json-simplify-41 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} + 1}{2} \]
      rational.json-simplify-1 [=>] 54.3	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 2 \cdot i\right)}} + 1}{2} \]

   3. Applied egg-rr 54.4

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

   4. Simplified 54.4

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

      [Start] 54.4	\[ \frac{\left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \left(\beta + \left(i + i\right)\right)} \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}\right)}{2} \]
      rational.json-simplify-43 [=>] 54.4	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\alpha + \left(\beta + \left(i + i\right)\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)} \cdot \left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right)\right)}}{2} \]
      rational.json-simplify-1 [=>] 54.4	\[ \frac{\frac{\color{blue}{\beta + \alpha}}{\alpha + \left(\beta + \left(i + i\right)\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)} \cdot \left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right)\right)}{2} \]
      rational.json-simplify-41 [=>] 54.4	\[ \frac{\frac{\beta + \alpha}{\color{blue}{\beta + \left(\left(i + i\right) + \alpha\right)}} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)} \cdot \left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right)\right)}{2} \]
      rational.json-simplify-1 [=>] 54.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\left(\left(i + i\right) + 2\right)}\right)} \cdot \left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right)\right)}{2} \]
      rational.json-simplify-41 [=>] 54.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \color{blue}{\left(\left(i + i\right) + \left(2 + \beta\right)\right)}} \cdot \left(\frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta} + 1\right)\right)}{2} \]
      rational.json-simplify-1 [=>] 54.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \color{blue}{\left(1 + \frac{\alpha + \left(\beta + \left(2 + \left(i + i\right)\right)\right)}{\beta - \alpha} \cdot \frac{\alpha + \left(\beta + \left(i + i\right)\right)}{\alpha + \beta}\right)}\right)}{2} \]

   5. Taylor expanded in alpha around inf 5.4

      \[\leadsto \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \color{blue}{\frac{-1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) - \left(\beta + -1 \cdot \beta\right)}{\alpha}}\right)}{2} \]

   6. Simplified 5.4

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

      [Start] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{-1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) - \left(\beta + -1 \cdot \beta\right)}{\alpha}\right)}{2} \]
      rational.json-simplify-2 [=>] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\color{blue}{\left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) \cdot -1} - \left(\beta + -1 \cdot \beta\right)}{\alpha}\right)}{2} \]
      rational.json-simplify-9 [=>] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\color{blue}{\left(-\left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)} - \left(\beta + -1 \cdot \beta\right)}{\alpha}\right)}{2} \]
      rational.json-simplify-2 [=>] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\left(-\left(\color{blue}{i \cdot 4} + \left(2 + 2 \cdot \beta\right)\right)\right) - \left(\beta + -1 \cdot \beta\right)}{\alpha}\right)}{2} \]
      rational.json-simplify-2 [=>] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\left(-\left(i \cdot 4 + \left(2 + 2 \cdot \beta\right)\right)\right) - \left(\beta + \color{blue}{\beta \cdot -1}\right)}{\alpha}\right)}{2} \]
      rational.json-simplify-8 [<=] 5.4	\[ \frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\left(-\left(i \cdot 4 + \left(2 + 2 \cdot \beta\right)\right)\right) - \left(\beta + \color{blue}{\left(-\beta\right)}\right)}{\alpha}\right)}{2} \]

   if -0.97999999999999998 < (/.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 13.1

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

   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] 13.1	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      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 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\beta + \left(\left(i + i\right) + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\alpha + \left(\left(i + i\right) + \left(2 + \beta\right)\right)} \cdot \frac{\left(-\left(i \cdot 4 + \left(2 + 2 \cdot \beta\right)\right)\right) - \left(\beta + \left(-\beta\right)\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	3524

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.9996:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternative 2
Error	7.1
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{2 \cdot i + \beta}}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	14.4
Cost	1364

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + \left(2 + \alpha\right)} + 1}{2}\\ t_1 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq -4.4 \cdot 10^{-263}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{-285}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.6 \cdot 10^{+167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.0
Cost	1236

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ t_1 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq -7 \cdot 10^{-263}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-286}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.8 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.12 \cdot 10^{+166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.75 \cdot 10^{+217}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.5
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 6.6 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	7.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	14.7
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq -5.4 \cdot 10^{-263}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{-285}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.5 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	7.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	15.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10
Error	18.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+116}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	25.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023068 
(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))