Octave 3.8, jcobi/4

Average Accuracy: 15.8% → 97.0%

Time: 33.2s

Precision: binary64

Cost: 27392

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

Math

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

↓

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

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

Derivation

1. Initial program 13.4%

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

2. Taylor expanded in alpha around 0 15.2%

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

3. Simplified 32.7%

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

   [Start] 15.2	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 32.7	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 32.7	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [=>] 32.7	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 88.3%

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

   [Start] 32.7	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   add-sqr-sqrt [=>] 32.7	\[ \frac{\color{blue}{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}} \cdot \sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   difference-of-sqr-1 [=>] 32.7	\[ \frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}} \cdot \sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
   times-frac [=>] 32.9	\[ \color{blue}{\frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

5. Simplified 88.3%

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

   [Start] 88.3	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
   associate-/l/ [=>] 88.3	\[ \color{blue}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
   +-commutative [=>] 88.3	\[ \frac{i}{\color{blue}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
   associate-/l/ [=>] 88.3	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \color{blue}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}} \]
   +-commutative [=>] 88.3	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\color{blue}{\left(-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]

6. Applied egg-rr 80.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{i}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)} \cdot \frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}\right)} - 1} \]
   Proof

   [Start] 88.3	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\left(-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]
   expm1-log1p-u [=>] 88.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\left(-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}\right)\right)} \]
   expm1-udef [=>] 80.4	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\left(-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}\right)} - 1} \]

7. Simplified 88.3%

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

   [Start] 80.4	\[ e^{\mathsf{log1p}\left(\frac{\frac{i}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)} \cdot \frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}\right)} - 1 \]
   expm1-def [=>] 88.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{i}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)} \cdot \frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}\right)\right)} \]
   expm1-log1p [=>] 88.3	\[ \color{blue}{\frac{\frac{i}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)} \cdot \frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}} \]
   associate-/l* [=>] 88.3	\[ \color{blue}{\frac{\frac{i}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)}}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}}} \]
   +-commutative [=>] 88.3	\[ \frac{\frac{i}{1 + \color{blue}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right) + -1}}} \]
   associate-+l+ [=>] 88.3	\[ \frac{\frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}}}} \]

8. Applied egg-rr 84.9%

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

   [Start] 88.3	\[ \frac{\frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}}} \]
   associate-/l/ [=>] 85.1	\[ \color{blue}{\frac{i}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}} \cdot \left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right)}} \]
   div-inv [=>] 84.9	\[ \color{blue}{i \cdot \frac{1}{\frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}} \cdot \left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right)}} \]
   div-inv [=>] 84.9	\[ i \cdot \frac{1}{\color{blue}{\left({\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{1}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}}\right)} \cdot \left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right)} \]
   clear-num [<=] 84.9	\[ i \cdot \frac{1}{\left({\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \color{blue}{\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}}\right) \cdot \left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right)} \]
   associate-*l* [=>] 84.9	\[ i \cdot \frac{1}{\color{blue}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i} \cdot \left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right)\right)}} \]
   +-commutative [=>] 84.9	\[ i \cdot \frac{1}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i} \cdot \color{blue}{\left(\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 1\right)}\right)} \]
   +-commutative [=>] 84.9	\[ i \cdot \frac{1}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i} \cdot \left(\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)} + 1\right)\right)} \]
   associate-+l+ [=>] 84.9	\[ i \cdot \frac{1}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i} \cdot \color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)}\right)} \]

9. Simplified 88.3%

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

   [Start] 84.9	\[ i \cdot \frac{1}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \left(\frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)\right)} \]
   associate-*r* [=>] 84.9	\[ i \cdot \frac{1}{\color{blue}{\left({\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)}} \]
   associate-/l/ [<=] 85.6	\[ i \cdot \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}}} \]
   associate-*r/ [=>] 88.2	\[ \color{blue}{\frac{i \cdot \frac{1}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}}} \]
   associate-*r/ [=>] 88.3	\[ \frac{\color{blue}{\frac{i \cdot 1}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]
   *-rgt-identity [=>] 88.3	\[ \frac{\frac{\color{blue}{i}}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]
   +-commutative [<=] 88.3	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{\left(1 + \alpha\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]
   +-commutative [=>] 88.3	\[ \frac{\frac{i}{\color{blue}{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]
   +-commutative [=>] 88.3	\[ \frac{\frac{i}{\color{blue}{\left(\alpha + 1\right)} + \mathsf{fma}\left(i, 2, \beta\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]
   +-commutative [<=] 88.3	\[ \frac{\frac{i}{\left(\alpha + 1\right) + \mathsf{fma}\left(i, 2, \beta\right)}}{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\color{blue}{\beta + i}}\right)}^{2} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}{i}} \]

10. Final simplification 88.3%

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

Alternatives

Alternative 1
Accuracy	96.9%
Cost	20992

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

Alternative 2
Accuracy	83.9%
Cost	16204

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + 2 \cdot \beta}{i}\right) - \frac{\beta}{i} \cdot \frac{\beta \cdot 0.03125}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{elif}\;\beta \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta + \left(1 + \left(\alpha + i \cdot \left(\left(1 + 2 \cdot \frac{\beta + \left(\alpha + 1\right)}{\beta}\right) + \left(\frac{-1}{\beta} - \frac{\alpha}{\beta}\right)\right)\right)\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	15180

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + 2 \cdot \beta}{i}\right) - \frac{\beta}{i} \cdot \frac{\beta \cdot 0.03125}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)} \cdot \frac{i}{\left(\left(\beta + \left(\alpha + 1\right)\right) + i \cdot 4\right) - i}\\ \end{array} \]

Alternative 4
Accuracy	84.6%
Cost	8968

\[\begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + 2 \cdot \beta}{i}\right) - \frac{\beta}{i} \cdot \frac{\beta \cdot 0.03125}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{\left(i + \beta\right) \cdot \left(i + \beta\right)}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta} + \frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	84.5%
Cost	2116

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + 2 \cdot \beta}{i}\right) - \frac{\beta}{i} \cdot \frac{\beta \cdot 0.03125}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{i + \alpha}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta} + \frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	84.8%
Cost	1356

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{i + \alpha}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta} + \frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	84.7%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+150}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	84.8%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+148}:\\ \;\;\;\;\frac{i + \alpha}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	81.6%
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.35 \cdot 10^{+150} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 10
Accuracy	81.7%
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	81.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 12
Accuracy	81.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.66 \cdot 10^{+149}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 13
Accuracy	74.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+255}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	75.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+210}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 15
Accuracy	70.6%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023160 
(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)))