Octave 3.8, jcobi/2

Average Error: 24.0 → 1.3

Time: 21.4s

Precision: binary64

Cost: 3524

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.995:\\ \;\;\;\;\frac{\beta - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{\frac{\beta - \alpha}{-2 \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \beta}}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)}\\ \end{array} \]

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)) -0.995)
     (/ (- beta (* 0.5 (+ -2.0 (* i -4.0)))) alpha)
     (-
      0.5
      (/
       (/
        (- beta alpha)
        (* -2.0 (/ (+ (+ alpha beta) (+ i i)) (+ alpha beta))))
       (+ alpha (+ (+ i i) (+ beta 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)) <= -0.995) {
		tmp = (beta - (0.5 * (-2.0 + (i * -4.0)))) / alpha;
	} else {
		tmp = 0.5 - (((beta - alpha) / (-2.0 * (((alpha + beta) + (i + i)) / (alpha + beta)))) / (alpha + ((i + i) + (beta + 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)) <= (-0.995d0)) then
        tmp = (beta - (0.5d0 * ((-2.0d0) + (i * (-4.0d0))))) / alpha
    else
        tmp = 0.5d0 - (((beta - alpha) / ((-2.0d0) * (((alpha + beta) + (i + i)) / (alpha + beta)))) / (alpha + ((i + i) + (beta + 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)) <= -0.995) {
		tmp = (beta - (0.5 * (-2.0 + (i * -4.0)))) / alpha;
	} else {
		tmp = 0.5 - (((beta - alpha) / (-2.0 * (((alpha + beta) + (i + i)) / (alpha + beta)))) / (alpha + ((i + i) + (beta + 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)) <= -0.995:
		tmp = (beta - (0.5 * (-2.0 + (i * -4.0)))) / alpha
	else:
		tmp = 0.5 - (((beta - alpha) / (-2.0 * (((alpha + beta) + (i + i)) / (alpha + beta)))) / (alpha + ((i + i) + (beta + 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)) <= -0.995)
		tmp = Float64(Float64(beta - Float64(0.5 * Float64(-2.0 + Float64(i * -4.0)))) / alpha);
	else
		tmp = Float64(0.5 - Float64(Float64(Float64(beta - alpha) / Float64(-2.0 * Float64(Float64(Float64(alpha + beta) + Float64(i + i)) / Float64(alpha + beta)))) / Float64(alpha + Float64(Float64(i + i) + Float64(beta + 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)) <= -0.995)
		tmp = (beta - (0.5 * (-2.0 + (i * -4.0)))) / alpha;
	else
		tmp = 0.5 - (((beta - alpha) / (-2.0 * (((alpha + beta) + (i + i)) / (alpha + beta)))) / (alpha + ((i + i) + (beta + 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], -0.995], N[(N[(beta - N[(0.5 * N[(-2.0 + N[(i * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(0.5 - N[(N[(N[(beta - alpha), $MachinePrecision] / N[(-2.0 * N[(N[(N[(alpha + beta), $MachinePrecision] + N[(i + i), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(N[(i + i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.994999999999999996

   1. Initial program 61.7

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

   2. Simplified 54.3

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-78 [=>] 61.7	\[ \frac{\color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}}{2} \]
      rational.json-simplify-23 [=>] 61.7	\[ \color{blue}{\frac{-1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]
      rational.json-simplify-31 [=>] 61.7	\[ \color{blue}{\frac{-1}{-2} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]
      metadata-eval [=>] 61.7	\[ \frac{-1}{\color{blue}{-2}} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      metadata-eval [=>] 61.7	\[ \color{blue}{0.5} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      rational.json-simplify-19 [=>] 54.3	\[ 0.5 - \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      rational.json-simplify-19 [=>] 54.3	\[ 0.5 - \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{-2} \]
      rational.json-simplify-39 [=>] 54.3	\[ 0.5 - \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{-2} \]
      rational.json-simplify-19 [=>] 54.3	\[ 0.5 - \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]

   3. Taylor expanded in alpha around inf 5.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}} \]

   4. Simplified 5.8

      \[\leadsto \color{blue}{\frac{\left(\beta + \left(\beta \cdot -1 - -1 \cdot \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)\right)\right) \cdot 0.5}{\alpha}} \]
      Proof

      [Start] 5.8	\[ 0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha} \]
      rational.json-simplify-20 [=>] 5.8	\[ \color{blue}{\frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot 0.5}{\alpha}} \]
      rational.json-simplify-13 [=>] 5.8	\[ \frac{\color{blue}{\left(\beta + \left(-1 \cdot \beta - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)\right)} \cdot 0.5}{\alpha} \]
      rational.json-simplify-39 [=>] 5.8	\[ \frac{\left(\beta + \left(\color{blue}{\beta \cdot -1} - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)\right) \cdot 0.5}{\alpha} \]
      rational.json-simplify-11 [=>] 5.8	\[ \frac{\left(\beta + \left(\beta \cdot -1 - -1 \cdot \color{blue}{\left(2 + \left(4 \cdot i + 2 \cdot \beta\right)\right)}\right)\right) \cdot 0.5}{\alpha} \]
      rational.json-simplify-39 [=>] 5.8	\[ \frac{\left(\beta + \left(\beta \cdot -1 - -1 \cdot \left(2 + \left(\color{blue}{i \cdot 4} + 2 \cdot \beta\right)\right)\right)\right) \cdot 0.5}{\alpha} \]
      rational.json-simplify-39 [=>] 5.8	\[ \frac{\left(\beta + \left(\beta \cdot -1 - -1 \cdot \left(2 + \left(i \cdot 4 + \color{blue}{\beta \cdot 2}\right)\right)\right)\right) \cdot 0.5}{\alpha} \]

   5. Applied egg-rr 5.9

      \[\leadsto \color{blue}{\left(\beta + \left(\beta - \left(-2 + i \cdot -4\right)\right)\right) \cdot \frac{0.5}{\alpha} + 0} \]

   6. Simplified 5.8

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

      [Start] 5.9	\[ \left(\beta + \left(\beta - \left(-2 + i \cdot -4\right)\right)\right) \cdot \frac{0.5}{\alpha} + 0 \]
      rational.json-simplify-65 [=>] 5.9	\[ \color{blue}{\left(\beta + \left(\beta - \left(-2 + i \cdot -4\right)\right)\right) \cdot \frac{0.5}{\alpha}} \]
      rational.json-simplify-20 [=>] 5.8	\[ \color{blue}{\frac{0.5 \cdot \left(\beta + \left(\beta - \left(-2 + i \cdot -4\right)\right)\right)}{\alpha}} \]
      rational.json-simplify-14 [=>] 5.8	\[ \frac{0.5 \cdot \color{blue}{\left(\left(\beta + \beta\right) - \left(-2 + i \cdot -4\right)\right)}}{\alpha} \]
      rational.json-simplify-34 [=>] 5.8	\[ \frac{\color{blue}{\left(\beta + \beta\right) \cdot 0.5 - 0.5 \cdot \left(-2 + i \cdot -4\right)}}{\alpha} \]
      metadata-eval [<=] 5.8	\[ \frac{\left(\beta + \beta\right) \cdot \color{blue}{\frac{0.5}{1}} - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha} \]
      rational.json-simplify-55 [<=] 5.8	\[ \frac{\color{blue}{\frac{\beta}{1}} - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha} \]
      rational.json-simplify-66 [=>] 5.8	\[ \frac{\color{blue}{\beta} - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha} \]

   if -0.994999999999999996 < (/.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

      \[\leadsto \color{blue}{0.5 - \frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)} \cdot \frac{\frac{\beta - \alpha}{\alpha + \left(2 + \left(\beta + 2 \cdot i\right)\right)}}{-2}} \]
      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-78 [=>] 12.9	\[ \frac{\color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - -1}}{2} \]
      rational.json-simplify-23 [=>] 12.9	\[ \color{blue}{\frac{-1 - \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]
      rational.json-simplify-31 [=>] 12.9	\[ \color{blue}{\frac{-1}{-2} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]
      metadata-eval [=>] 12.9	\[ \frac{-1}{\color{blue}{-2}} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      metadata-eval [=>] 12.9	\[ \color{blue}{0.5} - \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      rational.json-simplify-19 [=>] 0.0	\[ 0.5 - \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2} \]
      rational.json-simplify-19 [=>] 0.0	\[ 0.5 - \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{-2} \]
      rational.json-simplify-39 [=>] 0.0	\[ 0.5 - \frac{\color{blue}{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{-2} \]
      rational.json-simplify-19 [=>] 0.0	\[ 0.5 - \color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{-2}} \]

   3. Applied egg-rr 0.0

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\beta - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \frac{\frac{\beta - \alpha}{-2 \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \beta}}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	1988

\[\begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+133}:\\ \;\;\;\;0.5 - \frac{\alpha + \beta}{\alpha + t_0} \cdot \frac{\frac{\beta - \alpha}{\alpha + \left(2 + t_0\right)}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \end{array} \]

Alternative 2
Error	6.7
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;0.5 - \frac{\frac{\beta - \alpha}{-2 \cdot \left(1 + \frac{i}{\frac{\beta}{2}}\right)}}{\alpha + \left(\left(i + i\right) + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \end{array} \]

Alternative 3
Error	6.9
Cost	1412

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.4 \cdot 10^{+132}:\\ \;\;\;\;0.5 - \frac{-\frac{\beta}{\beta + \left(i + \left(2 + i\right)\right)}}{2 + \frac{i}{\frac{\beta}{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \end{array} \]

Alternative 4
Error	7.0
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.8 \cdot 10^{+132}:\\ \;\;\;\;0.5 - \frac{\beta}{\left(i + \left(\beta + \left(2 + i\right)\right)\right) \cdot \left(\frac{-4}{\frac{\beta}{i}} + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \end{array} \]

Alternative 5
Error	10.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+88}:\\ \;\;\;\;0.5 - \frac{\beta \cdot -0.5}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - 0.5 \cdot \left(-2 + i \cdot -4\right)}{\alpha}\\ \end{array} \]

Alternative 6
Error	13.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7 \cdot 10^{+132}:\\ \;\;\;\;0.5 - \frac{\beta \cdot -0.5}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{4 \cdot i + 2}{\alpha}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+113}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	24.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023066 
(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))