Octave 3.8, jcobi/4

Average Error: 53.9 → 2.0

Time: 24.5s

Precision: binary64

Cost: 27264

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

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

Math

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

↓

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

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

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

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

Derivation

1. Initial program 53.9

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

2. Taylor expanded in alpha around 0 54.0

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

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] 54.0	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. Taylor expanded in alpha around 0 39.7

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

6. Simplified 2.0

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

   [Start] 39.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \left(\beta + i\right)}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
   +-commutative [<=] 39.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \color{blue}{\left(i + \beta\right)}}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
   times-frac [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \color{blue}{\left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right)} \]
   +-commutative [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{i + \beta}{\color{blue}{2 \cdot i + \beta}}\right) \]
   *-commutative [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{i + \beta}{\color{blue}{i \cdot 2} + \beta}\right) \]
   fma-udef [<=] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{i + \beta}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}\right) \]
   associate-*r/ [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \color{blue}{\frac{\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)}} \]
   sub-neg [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\color{blue}{\left(\beta + 2 \cdot i\right) + \left(-1\right)}} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]
   +-commutative [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\color{blue}{\left(2 \cdot i + \beta\right)} + \left(-1\right)} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]
   *-commutative [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\left(\color{blue}{i \cdot 2} + \beta\right) + \left(-1\right)} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]
   fma-udef [<=] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \left(-1\right)} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]
   metadata-eval [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{-1}} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]
   +-commutative [=>] 2.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)} \]

7. Applied egg-rr 38.6

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

8. Simplified 2.0

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

   [Start] 38.6	\[ \frac{i}{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)\right) \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)\right)} \]
   associate-*r* [=>] 38.6	\[ \frac{i}{\color{blue}{\left(\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}} \]
   associate-*r* [<=] 38.6	\[ \frac{i}{\color{blue}{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right)\right)} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)} \]
   *-commutative [<=] 38.6	\[ \frac{i}{\color{blue}{\left(\left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)} \]
   associate-/l/ [<=] 37.9	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \]
   associate-*r/ [=>] 38.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\frac{\left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \]
   *-commutative [<=] 38.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right)}}{i + \beta}} \]
   associate-/l* [<=] 38.6	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}\right)}} \]
   associate-/r* [=>] 37.9	\[ \color{blue}{\frac{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)}}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}}} \]
   associate-*r/ [<=] 37.9	\[ \frac{\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}} \]
   associate-/l* [=>] 37.9	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}}} \]
   associate-*r/ [=>] 39.7	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\color{blue}{\frac{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \beta\right)}}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \]
   *-commutative [<=] 39.7	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\frac{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}}{i \cdot \left(i + \beta\right)}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \]
   *-commutative [=>] 39.7	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}{\color{blue}{\left(i + \beta\right) \cdot i}}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \]
   associate-/r* [=>] 36.2	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}{i + \beta}}{i}}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \]
   associate-*l/ [<=] 2.0	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}}{i}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \]
   associate-/r* [<=] 2.0	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\frac{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}{i \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}}} \]
   associate-/l* [<=] 2.0	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \left(i \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}} \]
   times-frac [=>] 2.0	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}} \]
   associate-*l/ [=>] 2.0	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \frac{i \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \]
   associate-*r/ [=>] 37.8	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \frac{\color{blue}{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*l/ [<=] 2.0	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \frac{\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*l/ [<=] 5.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \color{blue}{\left(\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(i + \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-/r/ [<=] 2.0	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}{i + \beta}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-/l/ [=>] 2.1	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}{i + \beta} \cdot \mathsf{fma}\left(i, 2, \beta\right)}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*r/ [=>] 2.0	\[ \frac{\color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}{i + \beta} \cdot \mathsf{fma}\left(i, 2, \beta\right)}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*l/ [=>] 33.2	\[ \frac{\frac{\color{blue}{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}{i + \beta} \cdot \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*l/ [=>] 37.2	\[ \frac{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\frac{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   *-commutative [<=] 37.2	\[ \frac{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\frac{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-*l/ [<=] 33.2	\[ \frac{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \]
   associate-/r* [<=] 33.2	\[ \color{blue}{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \]
   *-commutative [=>] 33.2	\[ \frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)\right)}} \]
   associate-*r* [=>] 33.2	\[ \frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)}} \]
   unpow2 [<=] 33.2	\[ \frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\color{blue}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \cdot \left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right)} \]
   associate-/r* [=>] 33.2	\[ \color{blue}{\frac{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}} \]
   associate-/l/ [=>] 33.2	\[ \color{blue}{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot {\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}} \]
   associate-/r* [=>] 33.2	\[ \color{blue}{\frac{\frac{\frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right) + -1}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}} \]
   associate-/l* [=>] 2.0	\[ \frac{\frac{\color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) + -1}{i}}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   fma-udef [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\color{blue}{\left(i \cdot 2 + \beta\right)} + -1}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   *-commutative [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\left(\color{blue}{2 \cdot i} + \beta\right) + -1}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   +-commutative [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\color{blue}{\left(\beta + 2 \cdot i\right)} + -1}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   metadata-eval [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\left(\beta + 2 \cdot i\right) + \color{blue}{\left(-1\right)}}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   sub-neg [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\color{blue}{\left(\beta + 2 \cdot i\right) - 1}}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   associate--l+ [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\color{blue}{\beta + \left(2 \cdot i - 1\right)}}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   fma-neg [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, -1\right)}}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   metadata-eval [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, \color{blue}{-1}\right)}{i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   +-commutative [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   fma-udef [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{1 + \color{blue}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   *-commutative [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{1 + \left(\color{blue}{2 \cdot i} + \left(\beta + \alpha\right)\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   associate-+r+ [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{1 + \color{blue}{\left(\left(2 \cdot i + \beta\right) + \alpha\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   *-commutative [=>] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{1 + \left(\left(\color{blue}{i \cdot 2} + \beta\right) + \alpha\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]
   fma-udef [<=] 2.0	\[ \frac{\frac{\frac{i}{\frac{\beta + \mathsf{fma}\left(2, i, -1\right)}{i}}}{1 + \left(\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \alpha\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}} \]

9. Final simplification 2.0

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

Alternatives

Alternative 1
Error	2.0
Cost	21312

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

Alternative 2
Error	9.2
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.3
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Error	9.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	15.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+210}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	10.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	10.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+167}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Error	16.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	57.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022356 
(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)))