Octave 3.8, jcobi/2

Average Error: 24.4 → 1.5

Time: 38.4s

Precision: binary64

Cost: 34372

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \left(\beta + \beta\right)\\ t_1 := \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := 2 + t_2\\ t_4 := \frac{\frac{\beta}{\alpha}}{\alpha}\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{t_3} \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(\left(\beta + 2 \cdot i\right) - \left(\left(-2 + i \cdot -2\right) - \beta\right)\right)\right) - \beta}{\alpha} + \left(\left(t_4 \cdot \left(\beta + 2\right) + \mathsf{fma}\left(i, \mathsf{fma}\left(2, \frac{\beta + 2}{\alpha \cdot \alpha}, \mathsf{fma}\left(4, t_1, \frac{-4}{\alpha} \cdot \frac{t_0}{\alpha}\right) + t_4 \cdot 6\right), t_1 \cdot t_0\right)\right) + \mathsf{fma}\left(-12, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, t_4 \cdot t_0\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_3} + 1}{2}\\ \end{array} \]

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 (+ (+ beta 2.0) (+ beta beta)))
        (t_1 (/ (- (- -2.0 beta) beta) (* alpha alpha)))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (+ 2.0 t_2))
        (t_4 (/ (/ beta alpha) alpha)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) t_3) -0.5)
     (/
      (+
       (/
        (- (+ beta (- (+ beta (* 2.0 i)) (- (+ -2.0 (* i -2.0)) beta))) beta)
        alpha)
       (+
        (+
         (* t_4 (+ beta 2.0))
         (fma
          i
          (fma
           2.0
           (/ (+ beta 2.0) (* alpha alpha))
           (+ (fma 4.0 t_1 (* (/ -4.0 alpha) (/ t_0 alpha))) (* t_4 6.0)))
          (* t_1 t_0)))
        (fma -12.0 (* (/ i alpha) (/ i alpha)) (* t_4 t_0))))
      2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta)))) t_3)
       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 = (beta + 2.0) + (beta + beta);
	double t_1 = ((-2.0 - beta) - beta) / (alpha * alpha);
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = 2.0 + t_2;
	double t_4 = (beta / alpha) / alpha;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / t_3) <= -0.5) {
		tmp = ((((beta + ((beta + (2.0 * i)) - ((-2.0 + (i * -2.0)) - beta))) - beta) / alpha) + (((t_4 * (beta + 2.0)) + fma(i, fma(2.0, ((beta + 2.0) / (alpha * alpha)), (fma(4.0, t_1, ((-4.0 / alpha) * (t_0 / alpha))) + (t_4 * 6.0))), (t_1 * t_0))) + fma(-12.0, ((i / alpha) * (i / alpha)), (t_4 * t_0)))) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / t_3) + 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(beta + 2.0) + Float64(beta + beta))
	t_1 = Float64(Float64(Float64(-2.0 - beta) - beta) / Float64(alpha * alpha))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(Float64(beta / alpha) / alpha)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / t_3) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + Float64(Float64(beta + Float64(2.0 * i)) - Float64(Float64(-2.0 + Float64(i * -2.0)) - beta))) - beta) / alpha) + Float64(Float64(Float64(t_4 * Float64(beta + 2.0)) + fma(i, fma(2.0, Float64(Float64(beta + 2.0) / Float64(alpha * alpha)), Float64(fma(4.0, t_1, Float64(Float64(-4.0 / alpha) * Float64(t_0 / alpha))) + Float64(t_4 * 6.0))), Float64(t_1 * t_0))) + fma(-12.0, Float64(Float64(i / alpha) * Float64(i / alpha)), Float64(t_4 * t_0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / t_3) + 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[(beta + 2.0), $MachinePrecision] + N[(beta + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(beta / alpha), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], -0.5], N[(N[(N[(N[(N[(beta + N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 + N[(i * -2.0), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(N[(t$95$4 * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * N[(2.0 * N[(N[(beta + 2.0), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * t$95$1 + N[(N[(-4.0 / alpha), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-12.0 * N[(N[(i / alpha), $MachinePrecision] * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $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 61.7

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

   2. Simplified 54.8

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 14.2

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

   4. Taylor expanded in i around 0 14.2

      \[\leadsto \frac{-1 \cdot \frac{\beta + -1 \cdot \left(\beta - \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)}{\alpha} + \color{blue}{\left(-12 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(\frac{\beta \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \left(i \cdot \left(2 \cdot \frac{\beta + 2}{{\alpha}^{2}} + \left(2 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{\beta}{{\alpha}^{2}} + \left(-4 \cdot \frac{\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)}{{\alpha}^{2}} + 4 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right)\right) + \left(\frac{\left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right) \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right)\right)}}{2} \]

   5. Simplified 6.2

      \[\leadsto \frac{-1 \cdot \frac{\beta + -1 \cdot \left(\beta - \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)}{\alpha} + \color{blue}{\left(\left(\frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(2 + \beta\right) + \mathsf{fma}\left(i, \mathsf{fma}\left(2, \frac{2 + \beta}{\alpha \cdot \alpha}, \mathsf{fma}\left(4, \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha}, \frac{-4}{\alpha} \cdot \frac{\left(\beta + \beta\right) - \left(-2 - \beta\right)}{\alpha}\right) + \frac{\frac{\beta}{\alpha}}{\alpha} \cdot 6\right), \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha} \cdot \left(\left(\beta + \beta\right) - \left(-2 - \beta\right)\right)\right)\right) + \mathsf{fma}\left(-12, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, \frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(\left(\beta + \beta\right) - \left(-2 - \beta\right)\right)\right)\right)}}{2} \]
      Proof

      [Start] 14.2

      \[ \frac{-1 \cdot \frac{\beta + -1 \cdot \left(\beta - \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)}{\alpha} + \left(-12 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(\frac{\beta \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \left(i \cdot \left(2 \cdot \frac{\beta + 2}{{\alpha}^{2}} + \left(2 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{\beta}{{\alpha}^{2}} + \left(-4 \cdot \frac{\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)}{{\alpha}^{2}} + 4 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right)\right) + \left(\frac{\left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right) \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right)\right)}{2} \]

      +-commutative [=>] 14.2

      \[ \frac{-1 \cdot \frac{\beta + -1 \cdot \left(\beta - \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)}{\alpha} + \color{blue}{\left(\left(\frac{\beta \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \left(i \cdot \left(2 \cdot \frac{\beta + 2}{{\alpha}^{2}} + \left(2 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{\beta}{{\alpha}^{2}} + \left(-4 \cdot \frac{\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)}{{\alpha}^{2}} + 4 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right)\right) + \left(\frac{\left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right) \cdot \left(\beta - \left(-1 \cdot \beta + -1 \cdot \left(\beta + 2\right)\right)\right)}{{\alpha}^{2}} + \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)\right)\right) + -12 \cdot \frac{{i}^{2}}{{\alpha}^{2}}\right)}}{2} \]

   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 13.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(\left(\beta + 2 \cdot i\right) - \left(\left(-2 + i \cdot -2\right) - \beta\right)\right)\right) - \beta}{\alpha} + \left(\left(\frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(\beta + 2\right) + \mathsf{fma}\left(i, \mathsf{fma}\left(2, \frac{\beta + 2}{\alpha \cdot \alpha}, \mathsf{fma}\left(4, \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha}, \frac{-4}{\alpha} \cdot \frac{\left(\beta + 2\right) + \left(\beta + \beta\right)}{\alpha}\right) + \frac{\frac{\beta}{\alpha}}{\alpha} \cdot 6\right), \frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha} \cdot \left(\left(\beta + 2\right) + \left(\beta + \beta\right)\right)\right)\right) + \mathsf{fma}\left(-12, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, \frac{\frac{\beta}{\alpha}}{\alpha} \cdot \left(\left(\beta + 2\right) + \left(\beta + \beta\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	9796

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \left(\beta + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ t_3 := \frac{\frac{\beta}{\alpha}}{\alpha}\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(\left(\beta + 2 \cdot i\right) - \left(\left(-2 + i \cdot -2\right) - \beta\right)\right)\right) - \beta}{\alpha} - \left(t_3 \cdot \left(-2 - \beta\right) - \left(\frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha} \cdot t_0 + t_3 \cdot t_0\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_2} + 1}{2}\\ \end{array} \]

Alternative 2
Error	2.6
Cost	5700

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \left(\beta + \beta\right)\\ t_1 := \beta + 2 \cdot i\\ t_2 := \frac{\frac{\beta}{\alpha}}{\alpha}\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_4 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_3}}{2 + t_3}\\ t_5 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;t_4 \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(t_1 - \left(\left(-2 + i \cdot -2\right) - \beta\right)\right)\right) - \beta}{\alpha} - \left(t_2 \cdot \left(-2 - \beta\right) - \left(\frac{\left(-2 - \beta\right) - \beta}{\alpha \cdot \alpha} \cdot t_0 + t_2 \cdot t_0\right)\right)}{2}\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\alpha + t_1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t_5}\right) + \alpha \cdot \frac{-1}{t_5}}{2}\\ \end{array} \]

Alternative 3
Error	2.6
Cost	5192

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1}\\ t_3 := \beta + 2 \cdot i\\ \mathbf{if}\;t_2 \leq -0.9999999:\\ \;\;\;\;\frac{\frac{t_3 - \left(\left(-2 + i \cdot -2\right) - \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\alpha + t_3\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t_0}\right) + \alpha \cdot \frac{-1}{t_0}}{2}\\ \end{array} \]

Alternative 4
Error	11.0
Cost	1864

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha}{\alpha + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) - \left(\left(-2 + i \cdot -2\right) - \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	16.5
Cost	1476

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;2 \cdot i \leq 10000000000:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t_0}\right) - \frac{\alpha}{t_0}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 10^{+60}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;2 \cdot i \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1356

\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10000000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 10^{+60}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;2 \cdot i \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Error	10.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) - \left(\left(-2 + i \cdot -2\right) - \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	16.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9
Error	18.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+99}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	25.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023059 
(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))