Octave 3.8, jcobi/4

Average Error: 54.1 → 9.5

Time: 22.2s

Precision: binary64

Cost: 22152

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := i + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right) \cdot \left(\left(i \cdot t_2\right) \cdot {t_0}^{-2}\right)}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t_0 + 1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

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 (fma i 2.0 (+ beta alpha)))
        (t_1 (+ (+ beta alpha) (* i 2.0)))
        (t_2 (+ i (+ beta alpha))))
   (if (<= beta 4.3478205419718985e+99)
     0.0625
     (if (<= beta 1.1649034665012227e+126)
       (/
        (* (fma i t_2 (* beta alpha)) (* (* i t_2) (pow t_0 -2.0)))
        (+ (* t_1 t_1) -1.0))
       (if (<= beta 1.3737296412870924e+150)
         0.0625
         (*
          (/ (+ i alpha) (+ t_0 1.0))
          (/ i (+ (+ beta alpha) (fma i 2.0 -1.0)))))))))

C

double code(double alpha, double beta, double i) {
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = i + (beta + alpha);
	double tmp;
	if (beta <= 4.3478205419718985e+99) {
		tmp = 0.0625;
	} else if (beta <= 1.1649034665012227e+126) {
		tmp = (fma(i, t_2, (beta * alpha)) * ((i * t_2) * pow(t_0, -2.0))) / ((t_1 * t_1) + -1.0);
	} else if (beta <= 1.3737296412870924e+150) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (t_0 + 1.0)) * (i / ((beta + alpha) + fma(i, 2.0, -1.0)));
	}
	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 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_2 = Float64(i + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.3478205419718985e+99)
		tmp = 0.0625;
	elseif (beta <= 1.1649034665012227e+126)
		tmp = Float64(Float64(fma(i, t_2, Float64(beta * alpha)) * Float64(Float64(i * t_2) * (t_0 ^ -2.0))) / Float64(Float64(t_1 * t_1) + -1.0));
	elseif (beta <= 1.3737296412870924e+150)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(t_0 + 1.0)) * Float64(i / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))));
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.3478205419718985e+99], 0.0625, If[LessEqual[beta, 1.1649034665012227e+126], N[(N[(N[(i * t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2), $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.3737296412870924e+150], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 4.3478205419718985e99 or 1.1649034665012227e126 < beta < 1.3737296412870924e150

   1. Initial program 49.0

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

   2. Simplified 33.2

      \[\leadsto \color{blue}{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 i (+.f64 beta (+.f64 i alpha))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta i) alpha))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta i)))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) i))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta i) alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta i))) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (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 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.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)))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (Rewrite<= associate-/r*_binary64 (/.f64 (+.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))))): 54 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (*.f64 i (+.f64 (+.f64 alpha beta) i)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 66 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_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))): 3 points increase in error, 7 points decrease in error

   3. Taylor expanded in beta around -inf 33.2

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

   4. Simplified 33.2

      \[\leadsto \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\color{blue}{\beta \cdot \beta + \left(\left(\alpha + i \cdot 2\right) \cdot \left(\left(\alpha + i \cdot 2\right) + \beta \cdot 2\right) + -1\right)}} \]
      Proof
      (+.f64 (*.f64 beta beta) (+.f64 (*.f64 (+.f64 alpha (*.f64 i 2)) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (+.f64 (*.f64 (+.f64 alpha (*.f64 i 2)) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (*.f64 (+.f64 alpha (Rewrite<= *-commutative_binary64 (*.f64 2 i))) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 (+.f64 alpha (Rewrite<= *-commutative_binary64 (*.f64 2 i))) (*.f64 beta 2))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 (+.f64 alpha (*.f64 2 i)) (Rewrite<= *-commutative_binary64 (*.f64 2 beta)))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 alpha (*.f64 2 i))) (*.f64 (*.f64 2 beta) (+.f64 alpha (*.f64 2 i))))) -1)): 0 points increase in error, 1 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2)) (*.f64 (*.f64 2 beta) (+.f64 alpha (*.f64 2 i)))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (+.f64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 beta (+.f64 alpha (*.f64 2 i)))))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (+.f64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2) (*.f64 2 (*.f64 beta (+.f64 alpha (*.f64 2 i))))) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2) (*.f64 2 (*.f64 beta (+.f64 alpha (*.f64 2 i))))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 beta 2) (+.f64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2) (*.f64 2 (*.f64 beta (+.f64 alpha (*.f64 2 i)))))) 1)): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in i around inf 4.7

      \[\leadsto \color{blue}{0.0625} \]

   if 4.3478205419718985e99 < beta < 1.1649034665012227e126

   1. Initial program 56.4

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

   2. Applied egg-rr 35.5

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

   if 1.3737296412870924e150 < beta

   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. Taylor expanded in beta around -inf 64.0

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

   3. Simplified 64.0

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\color{blue}{\beta \cdot \beta + \left(\alpha + i \cdot 2\right) \cdot \left(\left(\alpha + i \cdot 2\right) + \beta \cdot 2\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (+.f64 (*.f64 beta beta) (*.f64 (+.f64 alpha (*.f64 i 2)) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 (+.f64 alpha (*.f64 i 2)) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (*.f64 (+.f64 alpha (Rewrite<= *-commutative_binary64 (*.f64 2 i))) (+.f64 (+.f64 alpha (*.f64 i 2)) (*.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 (+.f64 alpha (Rewrite<= *-commutative_binary64 (*.f64 2 i))) (*.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 (+.f64 alpha (*.f64 2 i)) (Rewrite<= *-commutative_binary64 (*.f64 2 beta))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 alpha (*.f64 2 i)) (+.f64 alpha (*.f64 2 i))) (*.f64 (*.f64 2 beta) (+.f64 alpha (*.f64 2 i)))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2)) (*.f64 (*.f64 2 beta) (+.f64 alpha (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 beta 2) (+.f64 (pow.f64 (+.f64 alpha (*.f64 2 i)) 2) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 beta (+.f64 alpha (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in beta around inf 49.1

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

   5. Applied egg-rr 15.0

      \[\leadsto \color{blue}{\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right) \cdot \left(\left(i \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2}\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	14540

\[\begin{array}{l} t_0 := \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.5
Cost	972

\[\begin{array}{l} t_0 := \frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 4
Error	10.3
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{if}\;\beta \leq 4.3478205419718985 \cdot 10^{+99}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.1649034665012227 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 6
Error	10.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.3737296412870924 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 7
Error	16.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.132161392314007 \cdot 10^{+252}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	18.6
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022291 
(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)))