Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 84.6%

Time: 36.6s

Precision: binary64

Cost: 37572

• 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

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := 0.125 \cdot \frac{\beta}{i}\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t_3\\ t_5 := {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}\\ \mathbf{if}\;\frac{\frac{t_4 \cdot \left(t_4 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\frac{t_4}{t_5 + -1} \cdot \frac{\mathsf{fma}\left(i, t_3, \alpha \cdot \beta\right)}{t_5}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t_2\right) - t_2\\ \end{array} \]

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 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* 0.125 (/ beta i)))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3))
        (t_5 (pow (fma i 2.0 (+ alpha beta)) 2.0)))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (* (/ t_4 (+ t_5 -1.0)) (/ (fma i t_3 (* alpha beta)) t_5))
     (- (+ 0.0625 t_2) t_2))))

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 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = 0.125 * (beta / i);
	double t_3 = i + (alpha + beta);
	double t_4 = i * t_3;
	double t_5 = pow(fma(i, 2.0, (alpha + beta)), 2.0);
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (t_4 / (t_5 + -1.0)) * (fma(i, t_3, (alpha * beta)) / t_5);
	} else {
		tmp = (0.0625 + t_2) - t_2;
	}
	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 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(0.125 * Float64(beta / i))
	t_3 = Float64(i + Float64(alpha + beta))
	t_4 = Float64(i * t_3)
	t_5 = fma(i, 2.0, Float64(alpha + beta)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(t_4 / Float64(t_5 + -1.0)) * Float64(fma(i, t_3, Float64(alpha * beta)) / t_5));
	else
		tmp = Float64(Float64(0.0625 + t_2) - t_2);
	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[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$4 / N[(t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$2), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 46.3%

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

   2. Simplified 38.8%

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

      [Start] 46.3%	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 38.8%	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 38.8%	\[ \frac{\left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 38.8%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 38.8%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Applied egg-rr 99.7%

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

      [Start] 38.8%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)} \]
      times-frac [=>] 99.7%	\[ \color{blue}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right)} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)}} \]
      fma-udef [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      pow2 [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{{\left(\alpha + \left(\beta + i \cdot 2\right)\right)}^{2}} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      associate-+r+ [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\color{blue}{\left(\left(\alpha + \beta\right) + i \cdot 2\right)}}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      *-commutative [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\left(\left(\alpha + \beta\right) + \color{blue}{2 \cdot i}\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)}}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      *-commutative [<=] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\left(\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]
      fma-def [=>] 99.7%	\[ \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right)} \]

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.0%

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

   2. Simplified 3.4%

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

      [Start] 0.0%	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0%	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 0.0%	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      times-frac [=>] 0.0%	\[ \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Taylor expanded in i around inf 82.2%

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

   4. Taylor expanded in beta around inf 76.9%

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

   5. Taylor expanded in i around 0 76.9%

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

   6. Taylor expanded in beta around inf 78.0%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.6%
Cost	37572

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := 0.125 \cdot \frac{\beta}{i}\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t_3\\ t_5 := {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}\\ \mathbf{if}\;\frac{\frac{t_4 \cdot \left(t_4 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\frac{t_4}{t_5 + -1} \cdot \frac{\mathsf{fma}\left(i, t_3, \alpha \cdot \beta\right)}{t_5}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t_2\right) - t_2\\ \end{array} \]

Alternative 2
Accuracy	77.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+207}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Accuracy	83.7%
Cost	580

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

Alternative 4
Accuracy	71.5%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023271 
(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)))