Octave 3.8, jcobi/4

Average Error: 54.3 → 2.1

Time: 32.0s

Precision: binary64

Cost: 27840

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

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

Math

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

↓

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

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) (+ i beta))))
   (*
    (/ (/ i (* t_0 (+ 1.0 (+ (fma i 2.0 beta) alpha)))) t_0)
    (/ i (+ alpha (+ (fma i 2.0 beta) -1.0))))))

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) / (i + beta);
	return ((i / (t_0 * (1.0 + (fma(i, 2.0, beta) + alpha)))) / t_0) * (i / (alpha + (fma(i, 2.0, beta) + -1.0)));
}

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 = Float64(fma(i, 2.0, beta) / Float64(i + beta))
	return Float64(Float64(Float64(i / Float64(t_0 * Float64(1.0 + Float64(fma(i, 2.0, beta) + alpha)))) / t_0) * Float64(i / Float64(alpha + Float64(fma(i, 2.0, beta) + -1.0))))
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[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(i / N[(t$95$0 * N[(1.0 + N[(N[(i * 2.0 + beta), $MachinePrecision] + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(i / N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.3

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

2. Taylor expanded in alpha around 0 54.4

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

3. Simplified 42.1

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

   [Start] 54.4	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 42.1	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 42.1	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [=>] 42.1	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 2.0

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

5. Applied egg-rr 37.1

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

6. Simplified 2.1

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

   [Start] 37.1	\[ \frac{i \cdot i}{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)\right) \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)\right)} \]
   times-frac [=>] 2.1	\[ \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}} \]
   associate-*r/ [=>] 2.0	\[ \color{blue}{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)} \cdot i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}} \]
   times-frac [=>] 2.1	\[ \color{blue}{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}} \]
   +-commutative [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \color{blue}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   fma-udef [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \color{blue}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   *-commutative [<=] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \left(\color{blue}{2 \cdot i} + \left(\beta + \alpha\right)\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   associate-+r+ [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \color{blue}{\left(\left(2 \cdot i + \beta\right) + \alpha\right)}\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   *-commutative [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \left(\left(\color{blue}{i \cdot 2} + \beta\right) + \alpha\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   fma-udef [<=] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \alpha\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)} \]
   +-commutative [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\color{blue}{\left(\alpha + -1\right) + \mathsf{fma}\left(i, 2, \beta\right)}} \]
   associate-+l+ [=>] 2.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\color{blue}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}} \]

7. Final simplification 2.1

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

Alternatives

Alternative 1
Error	2.0
Cost	21440

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

Alternative 2
Error	9.9
Cost	15112

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0}\\ t_2 := t_1 \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ t_3 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;t_2 \cdot 0.25\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+151}:\\ \;\;\;\;\left(t_1 \cdot \frac{i + \beta}{t_3}\right) \cdot \frac{i \cdot \left(i + \beta\right)}{-1 + {t_3}^{2}}\\ \mathbf{elif}\;\beta \leq 4.1 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	14856

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ t_2 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \beta\right)\right)}^{2}}{-1 + t_2 \cdot t_2}\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Error	9.9
Cost	14796

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \frac{i}{t_1} \cdot \frac{i + \left(\beta + \alpha\right)}{t_1}\\ \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+132}:\\ \;\;\;\;t_2 \cdot 0.25\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	14276

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \frac{i}{t_1}\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+132}:\\ \;\;\;\;\left(t_2 \cdot \frac{i + \left(\beta + \alpha\right)}{t_1}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	9.9
Cost	8968

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 4.1 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	9.7
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	9.8
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+192}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + \alpha}\\ \end{array} \]

Alternative 9
Error	9.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + \alpha}\\ \end{array} \]

Alternative 10
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	16.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+251}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 12
Error	10.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 13
Error	57.2
Cost	64

\[0 \]

Alternative 14
Error	19.3
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023056 
(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)))