Octave 3.8, jcobi/4

Percentage Accurate: 15.3% → 83.9%

Time: 51.6s

Precision: binary64

Cost: 25604

• 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

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

↓

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ alpha (* i 2.0)))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (* i (+ alpha (+ i beta)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (*
      (/ t_4 (+ -1.0 (pow (+ alpha (+ beta (* i 2.0))) 2.0)))
      (/ (fma beta alpha t_4) (fma beta beta (* t_2 (+ t_2 (* beta 2.0))))))
     (+ (* -0.125 (/ beta i)) (+ 0.0625 (* (/ beta i) 0.125))))))

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 = alpha + (i * 2.0);
	double t_3 = i * (i + (alpha + beta));
	double t_4 = i * (alpha + (i + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (t_4 / (-1.0 + pow((alpha + (beta + (i * 2.0))), 2.0))) * (fma(beta, alpha, t_4) / fma(beta, beta, (t_2 * (t_2 + (beta * 2.0)))));
	} else {
		tmp = (-0.125 * (beta / i)) + (0.0625 + ((beta / i) * 0.125));
	}
	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(alpha + Float64(i * 2.0))
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(i * Float64(alpha + Float64(i + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(t_4 / Float64(-1.0 + (Float64(alpha + Float64(beta + Float64(i * 2.0))) ^ 2.0))) * Float64(fma(beta, alpha, t_4) / fma(beta, beta, Float64(t_2 * Float64(t_2 + Float64(beta * 2.0))))));
	else
		tmp = Float64(Float64(-0.125 * Float64(beta / i)) + Float64(0.0625 + Float64(Float64(beta / i) * 0.125)));
	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[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + 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[(-1.0 + N[Power[N[(alpha + N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha + t$95$4), $MachinePrecision] / N[(beta * beta + N[(t$95$2 * N[(t$95$2 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $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 48.8%

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

   2. Simplified 43.6%

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

      [Start] 48.8%	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 43.6%	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 43.6%	\[ \frac{\left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around -inf 43.6%

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

   4. Simplified 43.6%

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

      [Start] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left({\beta}^{2} + \left({\left(\alpha + 2 \cdot i\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right)} \]
      unpow2 [=>] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\color{blue}{\beta \cdot \beta} + \left({\left(\alpha + 2 \cdot i\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right)} \]
      *-commutative [<=] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + \color{blue}{i \cdot 2}\right)}^{2} + 2 \cdot \left(\beta \cdot \left(\alpha + 2 \cdot i\right)\right)\right)\right)} \]
      associate-*r* [=>] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \color{blue}{\left(2 \cdot \beta\right) \cdot \left(\alpha + 2 \cdot i\right)}\right)\right)} \]
      *-commutative [=>] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \color{blue}{\left(\beta \cdot 2\right)} \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      *-commutative [<=] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \left(\beta \cdot 2\right) \cdot \left(\alpha + \color{blue}{i \cdot 2}\right)\right)\right)} \]

   5. Applied egg-rr 99.7%

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

      [Start] 43.6%	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \left(\beta \cdot 2\right) \cdot \left(\alpha + i \cdot 2\right)\right)\right)} \]
      *-un-lft-identity [=>] 43.6%	\[ \color{blue}{1 \cdot \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right) \cdot \left(\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \left(\beta \cdot 2\right) \cdot \left(\alpha + i \cdot 2\right)\right)\right)}} \]
      times-frac [=>] 99.7%	\[ 1 \cdot \color{blue}{\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(\alpha + \left(\beta + i \cdot 2\right), \alpha + \left(\beta + i \cdot 2\right), -1\right)} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\beta \cdot \beta + \left({\left(\alpha + i \cdot 2\right)}^{2} + \left(\beta \cdot 2\right) \cdot \left(\alpha + i \cdot 2\right)\right)}\right)} \]

   6. Simplified 99.7%

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

      [Start] 99.7%	\[ 1 \cdot \left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)}\right) \]
      *-lft-identity [=>] 99.7%	\[ \color{blue}{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)}} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \color{blue}{\left(\left(\beta + \alpha\right) + i\right)}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + i\right)}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      associate-+l+ [=>] 99.7%	\[ \frac{i \cdot \color{blue}{\left(\alpha + \left(\beta + i\right)\right)}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \left(\alpha + \color{blue}{\left(i + \beta\right)}\right)}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2} + -1} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\color{blue}{-1 + {\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^{2}}} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{-1 + {\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot i\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      associate-+l+ [=>] 99.7%	\[ \frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{-1 + {\color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)}}^{2}} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]
      *-commutative [=>] 99.7%	\[ \frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{-1 + {\left(\alpha + \left(\beta + \color{blue}{i \cdot 2}\right)\right)}^{2}} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\mathsf{fma}\left(\beta, \beta, {\left(\alpha + 2 \cdot i\right)}^{2} + \beta \cdot \left(2 \cdot \left(\alpha + 2 \cdot i\right)\right)\right)} \]

   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 4.2%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]
      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 79.3%

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

   4. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right) \cdot \left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)\right) \cdot \left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)}} \]
      Step-by-step derivation

      [Start] 79.3%

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

      add-cbrt-cube [=>] 78.3%

      \[ \color{blue}{\sqrt[3]{\left(\left(\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right) \cdot \left(\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\right)}} \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(0.0625 + \frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right) \cdot -0.125}{i}\right) \cdot \left(\left(0.0625 + \frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right) \cdot -0.125}{i}\right) \cdot \left(0.0625 + \frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right) \cdot -0.125}{i}\right)\right)}} \]
      Step-by-step derivation

      [Start] 66.5%

      \[ \sqrt[3]{\left(\left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right) \cdot \left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)\right) \cdot \left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)} \]

      associate-*l* [=>] 66.5%

      \[ \sqrt[3]{\color{blue}{\left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right) \cdot \left(\left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right) \cdot \left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - 0.125 \cdot \frac{\beta + \alpha}{i}\right)\right)\right)}} \]

   6. Taylor expanded in alpha around 0 75.8%

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

4. Final simplification 83.1%

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

Alternatives

Alternative 1
Accuracy	83.9%
Cost	25604

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

Alternative 2
Accuracy	83.3%
Cost	24644

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\frac{i \cdot i}{-1 + {\left(\beta + i \cdot 2\right)}^{2}} \cdot \frac{{\left(i + \beta\right)}^{2}}{\mathsf{fma}\left(4, i \cdot \beta, \beta \cdot \beta + \left(i \cdot i\right) \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	12612

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

Alternative 4
Accuracy	80.1%
Cost	6852

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1}\\ \mathbf{if}\;t_3 \leq 0.0624999999:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \left(\beta \cdot \frac{1}{i + \alpha}\right)}\\ \end{array} \]

Alternative 6
Accuracy	77.4%
Cost	832

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

Alternative 7
Accuracy	76.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{1}{\beta} \cdot \frac{i}{\beta}\right)\\ \end{array} \]

Alternative 8
Accuracy	77.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{i + \alpha}{\beta}}}\\ \end{array} \]

Alternative 9
Accuracy	76.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \end{array} \]

Alternative 10
Accuracy	73.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+256}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 11
Accuracy	70.7%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023263 
(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)))