Octave 3.8, jcobi/2

Average Error: 24.1 → 1.5

Time: 21.3s

Precision: binary64

Cost: 70212

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

Math

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

↓

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

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 alpha) (- (- beta) (fma 2.0 i beta))))
        (t_1 (+ beta (- -2.0 (fma 2.0 i beta))))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.5)
     (/
      (fma
       -1.0
       (/ (- (fma -1.0 beta t_1) (fma 2.0 i beta)) alpha)
       (+
        (/ t_1 (/ alpha (/ (+ 2.0 (fma 2.0 i beta)) alpha)))
        (+ (/ -1.0 (/ t_0 t_1)) (/ (fma 2.0 i beta) t_0))))
      2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       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 * alpha) / (-beta - fma(2.0, i, beta));
	double t_1 = beta + (-2.0 - fma(2.0, i, beta));
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.5) {
		tmp = fma(-1.0, ((fma(-1.0, beta, t_1) - fma(2.0, i, beta)) / alpha), ((t_1 / (alpha / ((2.0 + fma(2.0, i, beta)) / alpha))) + ((-1.0 / (t_0 / t_1)) + (fma(2.0, i, beta) / t_0)))) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 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 * alpha) / Float64(Float64(-beta) - fma(2.0, i, beta)))
	t_1 = Float64(beta + Float64(-2.0 - fma(2.0, i, beta)))
	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(2.0 + t_2)) <= -0.5)
		tmp = Float64(fma(-1.0, Float64(Float64(fma(-1.0, beta, t_1) - fma(2.0, i, beta)) / alpha), Float64(Float64(t_1 / Float64(alpha / Float64(Float64(2.0 + fma(2.0, i, beta)) / alpha))) + Float64(Float64(-1.0 / Float64(t_0 / t_1)) + Float64(fma(2.0, i, beta) / t_0)))) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return 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 * alpha), $MachinePrecision] / N[((-beta) - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $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[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-1.0 * N[(N[(N[(-1.0 * beta + t$95$1), $MachinePrecision] - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(t$95$1 / N[(alpha / N[(N[(2.0 + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $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 61.6

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

   2. Simplified 53.1

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

      [Start] 61.6	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 61.6	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      times-frac [=>] 53.1	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      fma-def [=>] 53.1	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      fma-def [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}, 1\right)}{2} \]
      +-commutative [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}, 1\right)}{2} \]
      fma-def [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}, 1\right)}{2} \]

   3. Taylor expanded in alpha around -inf 15.1

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

   4. Simplified 6.7

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

      [Start] 15.1

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

      fma-def [=>] 15.1

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

   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 12.7

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

   2. Simplified 0.0

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

      [Start] 12.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 12.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      times-frac [=>] 0.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} + 1}{2} \]
      fma-def [=>] 0.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      fma-def [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}, 1\right)}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + 2 \cdot i\right)}}, 1\right)}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \color{blue}{\left(2 \cdot i + \beta\right)}}, 1\right)}{2} \]
      fma-def [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \color{blue}{\mathsf{fma}\left(2, i, \beta\right)}}, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.5
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 - \beta \cdot -2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	2.0
Cost	2756

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 - \beta \cdot -2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \left(\beta + i\right) \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(2 + 2 \cdot i\right)\right) + \left(\beta + 2 \cdot i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	14.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \left(\beta + i\right) \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3600000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \left(\beta + i\right) \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	16.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4500000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	17.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 10
Error	17.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+85}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	24.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023073 
(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))