Octave 3.8, jcobi/2

Average Error: 24.0 → 1.6

Time: 19.6s

Precision: binary64

Cost: 22340

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.95:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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 beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.95)
     (/ (+ (* 4.0 (/ i alpha)) (/ (+ beta (+ beta 2.0)) alpha)) 2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
       (/ (- beta alpha) (+ alpha (+ beta (fma 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) / (2.0 + t_0)) <= -0.95) {
		tmp = ((4.0 * (i / alpha)) + ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (beta + fma(2.0, i, alpha))), ((beta - alpha) / (alpha + (beta + fma(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(2.0 + t_0)) <= -0.95)
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))), Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 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 + 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[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.95], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $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)) < -0.94999999999999996

   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 53.0

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 61.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 53.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 53.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 53.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Taylor expanded in alpha around inf 6.9

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

   4. Taylor expanded in i around 0 6.9

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

   5. Simplified 6.9

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

      [Start] 6.9	\[ \frac{\left(\frac{\beta}{\alpha} + 4 \cdot \frac{i}{\alpha}\right) - -1 \cdot \frac{\beta + 2}{\alpha}}{2} \]
      +-commutative [=>] 6.9	\[ \frac{\color{blue}{\left(4 \cdot \frac{i}{\alpha} + \frac{\beta}{\alpha}\right)} - -1 \cdot \frac{\beta + 2}{\alpha}}{2} \]
      associate--l+ [=>] 6.9	\[ \frac{\color{blue}{4 \cdot \frac{i}{\alpha} + \left(\frac{\beta}{\alpha} - -1 \cdot \frac{\beta + 2}{\alpha}\right)}}{2} \]
      associate-*r/ [=>] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \left(\frac{\beta}{\alpha} - \color{blue}{\frac{-1 \cdot \left(\beta + 2\right)}{\alpha}}\right)}{2} \]
      div-sub [<=] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \color{blue}{\frac{\beta - -1 \cdot \left(\beta + 2\right)}{\alpha}}}{2} \]
      cancel-sign-sub-inv [=>] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \frac{\color{blue}{\beta + \left(--1\right) \cdot \left(\beta + 2\right)}}{\alpha}}{2} \]
      metadata-eval [=>] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \color{blue}{1} \cdot \left(\beta + 2\right)}{\alpha}}{2} \]
      *-commutative [<=] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \color{blue}{\left(\beta + 2\right) \cdot 1}}{\alpha}}{2} \]
      *-rgt-identity [=>] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \color{blue}{\left(\beta + 2\right)}}{\alpha}}{2} \]
      +-commutative [=>] 6.9	\[ \frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \color{blue}{\left(2 + \beta\right)}}{\alpha}}{2} \]

   if -0.94999999999999996 < (/.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

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

3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.95:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.9
Cost	11592

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ t_2 := \frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}\\ t_3 := \frac{t_2}{t_1}\\ \mathbf{if}\;t_3 \leq -0.95:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_2, \frac{1}{t_1}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 2
Error	2.9
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.95:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 3
Error	11.7
Cost	1492

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+59}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	11.2
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 10^{+59} \lor \neg \left(\alpha \leq 7.2 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]

Alternative 5
Error	11.1
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 2.55 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	13.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	17.0
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+24} \lor \neg \left(\alpha \leq 4.4 \cdot 10^{+58}\right) \land \alpha \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.45 \cdot 10^{+58} \lor \neg \left(\alpha \leq 1.65 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9
Error	13.5
Cost	972

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	13.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 7.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5.4 \cdot 10^{+96}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	17.6
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+50}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	24.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023187 
(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))