Octave 3.8, jcobi/2

Average Error: 23.8 → 2.0

Time: 10.5s

Precision: binary64

Cost: 3012

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(i \cdot 2 + \beta\right)}{\alpha + \left(\left(\beta + 2\right) + i \cdot 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]

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)) -2e-15)
     (/
      (/
       (+ 2.0 (* 2.0 (+ (* i 2.0) beta)))
       (+ alpha (+ (+ beta 2.0) (* i 2.0))))
      2.0)
     (/ (+ (/ beta (+ (+ beta (* 2.0 i)) 2.0)) 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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -2e-15) {
		tmp = ((2.0 + (2.0 * ((i * 2.0) + beta))) / (alpha + ((beta + 2.0) + (i * 2.0)))) / 2.0;
	} else {
		tmp = ((beta / ((beta + (2.0 * i)) + 2.0)) + 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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) <= (-2d-15)) then
        tmp = ((2.0d0 + (2.0d0 * ((i * 2.0d0) + beta))) / (alpha + ((beta + 2.0d0) + (i * 2.0d0)))) / 2.0d0
    else
        tmp = ((beta / ((beta + (2.0d0 * i)) + 2.0d0)) + 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)) <= -2e-15) {
		tmp = ((2.0 + (2.0 * ((i * 2.0) + beta))) / (alpha + ((beta + 2.0) + (i * 2.0)))) / 2.0;
	} else {
		tmp = ((beta / ((beta + (2.0 * i)) + 2.0)) + 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 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -2e-15:
		tmp = ((2.0 + (2.0 * ((i * 2.0) + beta))) / (alpha + ((beta + 2.0) + (i * 2.0)))) / 2.0
	else:
		tmp = ((beta / ((beta + (2.0 * i)) + 2.0)) + 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(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)) <= -2e-15)
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(Float64(i * 2.0) + beta))) / Float64(alpha + Float64(Float64(beta + 2.0) + Float64(i * 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / Float64(Float64(beta + Float64(2.0 * i)) + 2.0)) + 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 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -2e-15)
		tmp = ((2.0 + (2.0 * ((i * 2.0) + beta))) / (alpha + ((beta + 2.0) + (i * 2.0)))) / 2.0;
	else
		tmp = ((beta / ((beta + (2.0 * i)) + 2.0)) + 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[(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], -2e-15], N[(N[(N[(2.0 + N[(2.0 * N[(N[(i * 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(N[(beta + 2.0), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $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)) < -2.0000000000000002e-15

   1. Initial program 58.6

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

   2. Taylor expanded in alpha around inf 55.4

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

   3. Simplified 55.4

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

      [Start] 55.4	\[ \frac{\frac{\left(-1 \cdot \beta + \left(\beta + -1 \cdot \alpha\right)\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-33 [=>] 55.4	\[ \frac{\frac{\color{blue}{\left(\left(-1 \cdot \beta + \beta\right) + -1 \cdot \alpha\right)} - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [<=] 55.4	\[ \frac{\frac{\left(\color{blue}{\left(\beta + -1 \cdot \beta\right)} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-2 [=>] 55.4	\[ \frac{\frac{\left(\left(\beta + \color{blue}{\beta \cdot -1}\right) + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-19 [=>] 55.4	\[ \frac{\frac{\left(\color{blue}{\beta \cdot \left(1 + -1\right)} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      metadata-eval [=>] 55.4	\[ \frac{\frac{\left(\beta \cdot \color{blue}{0} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      metadata-eval [<=] 55.4	\[ \frac{\frac{\left(\beta \cdot \color{blue}{\left(-1 \cdot 0\right)} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-31 [<=] 55.4	\[ \frac{\frac{\left(\color{blue}{-1 \cdot \left(\beta \cdot 0\right)} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      metadata-eval [<=] 55.4	\[ \frac{\frac{\left(-1 \cdot \left(\beta \cdot \color{blue}{\left(1 + -1\right)}\right) + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-19 [<=] 55.4	\[ \frac{\frac{\left(-1 \cdot \color{blue}{\left(\beta + \beta \cdot -1\right)} + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-2 [<=] 55.4	\[ \frac{\frac{\left(-1 \cdot \left(\beta + \color{blue}{-1 \cdot \beta}\right) + -1 \cdot \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-40 [=>] 55.4	\[ \frac{\frac{\color{blue}{-\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \left(-1 \cdot \left(\beta + -1 \cdot \beta\right) + -1 \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 55.4	\[ \frac{\frac{-\left(-1 \cdot \left(\beta + 2 \cdot i\right) - \color{blue}{\left(-1 \cdot \alpha + -1 \cdot \left(\beta + -1 \cdot \beta\right)\right)}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-39 [=>] 55.4	\[ \frac{\frac{-\color{blue}{\left(\left(-1 \cdot \left(\beta + 2 \cdot i\right) - -1 \cdot \left(\beta + -1 \cdot \beta\right)\right) - -1 \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   4. Applied egg-rr 5.3

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

   5. Simplified 5.3

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

      [Start] 5.3	\[ \frac{\frac{2 + 2 \cdot \left(\beta + 2 \cdot i\right)}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}{2} \]
      rational.json-simplify-1 [=>] 5.3	\[ \frac{\frac{2 + 2 \cdot \color{blue}{\left(2 \cdot i + \beta\right)}}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}{2} \]
      rational.json-simplify-2 [=>] 5.3	\[ \frac{\frac{2 + 2 \cdot \left(\color{blue}{i \cdot 2} + \beta\right)}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)}}{2} \]
      rational.json-simplify-33 [=>] 5.3	\[ \frac{\frac{2 + 2 \cdot \left(i \cdot 2 + \beta\right)}{\alpha + \color{blue}{\left(\left(\beta + 2\right) + 2 \cdot i\right)}}}{2} \]
      rational.json-simplify-2 [=>] 5.3	\[ \frac{\frac{2 + 2 \cdot \left(i \cdot 2 + \beta\right)}{\alpha + \left(\left(\beta + 2\right) + \color{blue}{i \cdot 2}\right)}}{2} \]

   if -2.0000000000000002e-15 < (/.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 12.8

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

   2. Taylor expanded in beta around inf 0.9

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

   3. Taylor expanded in alpha around 0 0.9

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(\beta + 2 \cdot i\right)} + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(i \cdot 2 + \beta\right)}{\alpha + \left(\left(\beta + 2\right) + i \cdot 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.05 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.0
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 75000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 9 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{2}{\alpha - \left(-2 - 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 33500000:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{2}{\alpha - \left(-2 - 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 0.04:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\beta + \left(2 + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \left(i \cdot 2 + \beta\right) \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	5.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\beta + \left(2 + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 6
Error	15.4
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 4.6 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \frac{i}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	16.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.1 \cdot 10^{+226}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{i}{\alpha}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	588

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 72000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+134}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 10
Error	17.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 160000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	24.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023053 
(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))