Octave 3.8, jcobi/4

Average Error: 53.7 → 9.6

Time: 24.1s

Precision: binary64

Cost: 14797

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \frac{i}{t_0}\\ t_2 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;\left(t_1 \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+181} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+190}\right):\\ \;\;\;\;\left(\frac{i}{t_2} \cdot \frac{i + \left(\beta + \alpha\right)}{t_2}\right) \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot 0.5\right)\\ \end{array} \]

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 (+ beta (* i 2.0)))
        (t_1 (/ i t_0))
        (t_2 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 2.2e+134)
     (* (* t_1 (/ (+ beta i) t_0)) 0.25)
     (if (or (<= beta 1.15e+181) (not (<= beta 2.1e+190)))
       (* (* (/ i t_2) (/ (+ i (+ beta alpha)) t_2)) (/ (+ i alpha) beta))
       (* 0.25 (* t_1 0.5))))))

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 = beta + (i * 2.0);
	double t_1 = i / t_0;
	double t_2 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 2.2e+134) {
		tmp = (t_1 * ((beta + i) / t_0)) * 0.25;
	} else if ((beta <= 1.15e+181) || !(beta <= 2.1e+190)) {
		tmp = ((i / t_2) * ((i + (beta + alpha)) / t_2)) * ((i + alpha) / beta);
	} else {
		tmp = 0.25 * (t_1 * 0.5);
	}
	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(beta + Float64(i * 2.0))
	t_1 = Float64(i / t_0)
	t_2 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.2e+134)
		tmp = Float64(Float64(t_1 * Float64(Float64(beta + i) / t_0)) * 0.25);
	elseif ((beta <= 1.15e+181) || !(beta <= 2.1e+190))
		tmp = Float64(Float64(Float64(i / t_2) * Float64(Float64(i + Float64(beta + alpha)) / t_2)) * Float64(Float64(i + alpha) / beta));
	else
		tmp = Float64(0.25 * Float64(t_1 * 0.5));
	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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.2e+134], N[(N[(t$95$1 * N[(N[(beta + i), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], If[Or[LessEqual[beta, 1.15e+181], N[Not[LessEqual[beta, 2.1e+190]], $MachinePrecision]], N[(N[(N[(i / t$95$2), $MachinePrecision] * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.2e134

   1. Initial program 48.5

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

   2. Simplified 32.1

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (*.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)))): 0 points increase in error, 21 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.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))): 0 points increase in error, 23 points decrease in error

   3. Taylor expanded in i around inf 5.3

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \color{blue}{0.25} \]

   4. Taylor expanded in alpha around 0 5.3

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{\frac{\beta + i}{\beta + 2 \cdot i}}\right) \cdot 0.25 \]

   5. Taylor expanded in alpha around 0 5.3

      \[\leadsto \left(\color{blue}{\frac{i}{\beta + 2 \cdot i}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]

   6. Simplified 5.3

      \[\leadsto \left(\color{blue}{\frac{i}{\beta + i \cdot 2}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]
      Proof
      (*.f64 (*.f64 (/.f64 i (+.f64 beta (*.f64 i 2))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 2 i)))) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 2 i)))) 1/4): 2 points increase in error, 0 points decrease in error

   if 2.2e134 < beta < 1.1499999999999999e181 or 2.1000000000000001e190 < beta

   1. Initial program 63.8

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

   2. Simplified 54.2

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (*.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)))): 0 points increase in error, 21 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.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))): 0 points increase in error, 23 points decrease in error

   3. Taylor expanded in beta around inf 17.0

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

   if 1.1499999999999999e181 < beta < 2.1000000000000001e190

   1. Initial program 64.0

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

   2. Simplified 58.6

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (*.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)))): 0 points increase in error, 21 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.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))): 0 points increase in error, 23 points decrease in error

   3. Taylor expanded in i around inf 35.5

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \color{blue}{0.25} \]

   4. Taylor expanded in alpha around 0 35.5

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{\frac{\beta + i}{\beta + 2 \cdot i}}\right) \cdot 0.25 \]

   5. Taylor expanded in alpha around 0 35.5

      \[\leadsto \left(\color{blue}{\frac{i}{\beta + 2 \cdot i}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]

   6. Simplified 35.5

      \[\leadsto \left(\color{blue}{\frac{i}{\beta + i \cdot 2}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]
      Proof
      (*.f64 (*.f64 (/.f64 i (+.f64 beta (*.f64 i 2))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 2 i)))) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 2 i)))) 1/4): 2 points increase in error, 0 points decrease in error

   7. Taylor expanded in beta around 0 36.2

      \[\leadsto \left(\frac{i}{\beta + i \cdot 2} \cdot \color{blue}{0.5}\right) \cdot 0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+181} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+190}\right):\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	1348

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \frac{i}{t_0}\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;\left(t_1 \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\frac{\frac{\beta}{i + \alpha}}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+189}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2
Error	9.9
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+181}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+189}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\frac{\frac{\beta}{i + \alpha}}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+190}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\beta + i \cdot 2} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Error	9.9
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+181}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+181} \lor \neg \left(\beta \leq 3.3 \cdot 10^{+189}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 6
Error	16.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	19.1
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022343 
(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)))