Octave 3.8, jcobi/4

Average Error: 54.5 → 1.9

Time: 33.9s

Precision: binary64

Cost: 27712

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

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

Math

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

↓

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

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

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

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)
	return Float64(Float64(Float64(Float64(Float64(i / fma(i, 2.0, beta)) * Float64(i + beta)) / Float64(fma(i, 2.0, beta) + Float64(-1.0 + alpha))) / Float64(Float64(fma(i, 2.0, beta) + 1.0) / Float64(i + beta))) / Float64(fma(i, 2.0, beta) / i))
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_] := N[(N[(N[(N[(N[(i / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] + N[(-1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] + 1.0), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5

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

2. Taylor expanded in alpha around 0 54.6

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

3. Simplified 42.8

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

   [Start] 54.6	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 42.8	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 42.8	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [=>] 42.8	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

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. Taylor expanded in alpha around 0 40.4

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

6. Simplified 2.0

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

   [Start] 40.4	\[ \frac{i \cdot \left(\beta + i\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [<=] 40.4	\[ \frac{i \cdot \color{blue}{\left(i + \beta\right)}}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   times-frac [=>] 2.0	\[ \color{blue}{\left(\frac{i}{\beta + 2 \cdot i} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [=>] 2.0	\[ \left(\frac{i}{\color{blue}{2 \cdot i + \beta}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   *-commutative [=>] 2.0	\[ \left(\frac{i}{\color{blue}{i \cdot 2} + \beta} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   fma-udef [<=] 2.0	\[ \left(\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   associate-+r+ [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{\left(\beta + 1\right) + 2 \cdot i}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   *-commutative [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\left(\beta + 1\right) + \color{blue}{i \cdot 2}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]

7. Applied egg-rr 1.9

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

8. Final simplification 1.9

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

Alternatives

Alternative 1
Error	2.0
Cost	21632

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

Alternative 2
Error	2.0
Cost	21440

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

Alternative 3
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 4
Error	11.0
Cost	15564

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\left(\beta + 1\right) + i \cdot 2}\\ t_2 := \alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;t_1 \cdot \frac{\left(i + \beta\right) \cdot 0.5 + \beta \cdot -0.25}{t_2}\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+168}:\\ \;\;\;\;t_1 \cdot \frac{i}{t_2}\\ \mathbf{elif}\;\beta \leq 2.55 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{t_2} \cdot \left(\left(0.25 + \frac{\beta \cdot 0.25}{i}\right) + 0.0625 \cdot \frac{\left(\beta + \left(\beta + 1\right)\right) \cdot -2}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	15172

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\left(\beta + 1\right) + i \cdot 2}\\ t_2 := \alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+92}:\\ \;\;\;\;t_1 \cdot \frac{\left(i + \beta\right) \cdot 0.5 + \beta \cdot -0.25}{t_2}\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;t_1 \cdot \frac{i}{t_2}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+222}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	11.6
Cost	14797

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+168} \lor \neg \left(\beta \leq 2.7 \cdot 10^{+222}\right):\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 7
Error	11.7
Cost	14796

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+91}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{\left(i + \beta\right) + \alpha}}}{t_0 \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.6 \cdot 10^{+222}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	14796

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+168}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\left(\beta + 1\right) + i \cdot 2}\right) \cdot \frac{i}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	11.6
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0}\\ \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right)\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+168} \lor \neg \left(\beta \leq 3.2 \cdot 10^{+222}\right):\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 10
Error	11.6
Cost	7629

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+168} \lor \neg \left(\beta \leq 2.45 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 11
Error	11.8
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+168}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+222}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 12
Error	12.4
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \left(i \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 13
Error	12.4
Cost	973

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+168} \lor \neg \left(\beta \leq 2.45 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 14
Error	12.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+168}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 15
Error	12.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+168}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 16
Error	13.6
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+168} \lor \neg \left(\beta \leq 2.45 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 17
Error	13.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+93}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 18
Error	16.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 19
Error	18.4
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023034 
(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)))