Octave 3.8, jcobi/4

Average Error: 54.1 → 10.0

Time: 23.3s

Precision: binary64

Cost: 14541

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+192} \lor \neg \left(\beta \leq 9 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{\frac{i}{\frac{t_0 + -1}{i + \alpha}}}{t_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 4.8e+98)
     (* (* (/ i t_0) (/ (+ beta i) (+ beta (* i 2.0)))) 0.25)
     (if (or (<= beta 1.55e+192) (not (<= beta 9e+201)))
       (/ (/ i (/ (+ t_0 -1.0) (+ i alpha))) (+ t_0 1.0))
       (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125))))))

C

double code(double alpha, double beta, double i) {
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 4.8e+98) {
		tmp = ((i / t_0) * ((beta + i) / (beta + (i * 2.0)))) * 0.25;
	} else if ((beta <= 1.55e+192) || !(beta <= 9e+201)) {
		tmp = (i / ((t_0 + -1.0) / (i + alpha))) / (t_0 + 1.0);
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.8e+98)
		tmp = Float64(Float64(Float64(i / t_0) * Float64(Float64(beta + i) / Float64(beta + Float64(i * 2.0)))) * 0.25);
	elseif ((beta <= 1.55e+192) || !(beta <= 9e+201))
		tmp = Float64(Float64(i / Float64(Float64(t_0 + -1.0) / Float64(i + alpha))) / Float64(t_0 + 1.0));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.8e+98], N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], If[Or[LessEqual[beta, 1.55e+192], N[Not[LessEqual[beta, 9e+201]], $MachinePrecision]], N[(N[(i / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.7999999999999997e98

   1. Initial program 48.0

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

   2. Simplified 31.3

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

      [Start] 48.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 48.8	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 31.3	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 3.2

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \color{blue}{0.25} \]

   4. Taylor expanded in alpha around 0 3.2

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{\frac{\beta + i}{\beta + 2 \cdot i}}\right) \cdot 0.25 \]

   if 4.7999999999999997e98 < beta < 1.5499999999999999e192 or 9.0000000000000002e201 < beta

   1. Initial program 62.8

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

   2. Taylor expanded in beta around inf 46.0

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

   3. Applied egg-rr 34.4

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

   4. Simplified 18.9

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

      [Start] 34.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*l/ [=>] 34.4	\[ \color{blue}{\frac{1 \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}} \]
      associate-/l* [=>] 18.9	\[ \frac{1 \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 18.9	\[ \frac{1 \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 18.9	\[ \frac{1 \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \alpha}}}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \]

   if 1.5499999999999999e192 < beta < 9.0000000000000002e201

   1. Initial program 64.0

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

   2. Simplified 58.3

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 58.3	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 34.9

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   4. Taylor expanded in beta around inf 35.0

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(2 \cdot \frac{\beta}{i}\right)}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   5. Simplified 35.0

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      Proof

      [Start] 35.0	\[ \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      associate-*r/ [=>] 35.0	\[ \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   6. Taylor expanded in i around 0 35.0

      \[\leadsto \color{blue}{\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   7. Taylor expanded in beta around inf 34.9

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+192} \lor \neg \left(\beta \leq 9 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	14541

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+192} \lor \neg \left(\beta \leq 8.2 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{i + \alpha}{t_0 + 1} \cdot \frac{i}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 2
Error	10.8
Cost	7748

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\left(t_0 \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+192} \lor \neg \left(\beta \leq 8.2 \cdot 10^{+201}\right):\\ \;\;\;\;t_0 \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 3
Error	10.8
Cost	7629

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+192} \lor \neg \left(\beta \leq 8.2 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 4
Error	11.0
Cost	1228

\[\begin{array}{l} t_0 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+184}:\\ \;\;\;\;t_0 \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+201}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{i}{\beta + \alpha}\\ \end{array} \]

Alternative 5
Error	10.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + \alpha}\\ \end{array} \]

Alternative 6
Error	10.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+98}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	17.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+212}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+137}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+141}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	17.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+209}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	57.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023046 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))