Octave 3.8, jcobi/4

Average Error: 53.7 → 9.7

Time: 19.2s

Precision: binary64

Cost: 964

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta}{\frac{i}{0.125}}\\ \mathbf{if}\;\beta \leq 8.201056857348168 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

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 0.125))))
   (if (<= beta 8.201056857348168e+185)
     (- (+ 0.0625 t_0) t_0)
     (/ (/ (+ i alpha) beta) (/ beta i)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = beta / (i / 0.125);
	double tmp;
	if (beta <= 8.201056857348168e+185) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i)))) / ((((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i))) - 1.0d0)
end function

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta / (i / 0.125d0)
    if (beta <= 8.201056857348168d+185) then
        tmp = (0.0625d0 + t_0) - t_0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

public static 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);
}

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = beta / (i / 0.125);
	double tmp;
	if (beta <= 8.201056857348168e+185) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

def code(alpha, beta, 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)

↓

def code(alpha, beta, i):
	t_0 = beta / (i / 0.125)
	tmp = 0
	if beta <= 8.201056857348168e+185:
		tmp = (0.0625 + t_0) - t_0
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	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 / 0.125))
	tmp = 0.0
	if (beta <= 8.201056857348168e+185)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((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);
end

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = beta / (i / 0.125);
	tmp = 0.0;
	if (beta <= 8.201056857348168e+185)
		tmp = (0.0625 + t_0) - t_0;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = 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 / 0.125), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 8.201056857348168e+185], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.20105685734816829e185

   1. Initial program 50.5

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

   2. Simplified 35.3

      \[\leadsto \color{blue}{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)}} \]
      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))))): 56 points increase in error, 2 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)))))): 69 points increase in error, 8 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))): 4 points increase in error, 2 points decrease in error

   3. Taylor expanded in i around inf 8.3

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

   4. Taylor expanded in beta around inf 8.4

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

   5. Simplified 8.4

      \[\leadsto \left(0.0625 + \color{blue}{\frac{\beta}{\frac{i}{0.125}}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      Proof
      (/.f64 beta (/.f64 i 1/8)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 beta i) 1/8)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/8 (/.f64 beta i))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in beta around inf 8.3

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

   7. Simplified 8.3

      \[\leadsto \left(0.0625 + \frac{\beta}{\frac{i}{0.125}}\right) - \color{blue}{\frac{\beta}{\frac{i}{0.125}}} \]
      Proof
      (/.f64 beta (/.f64 i 1/8)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 beta i) 1/8)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/8 (/.f64 beta i))): 0 points increase in error, 0 points decrease in error

   if 8.20105685734816829e185 < 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. Simplified 56.8

      \[\leadsto \color{blue}{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)}} \]
      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))))): 56 points increase in error, 2 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)))))): 69 points increase in error, 8 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))): 4 points increase in error, 2 points decrease in error

   3. Taylor expanded in beta around inf 46.9

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

   4. Simplified 45.5

      \[\leadsto \color{blue}{\frac{i}{\beta \cdot \beta} \cdot \left(\alpha + i\right)} \]
      Proof
      (*.f64 (/.f64 i (*.f64 beta beta)) (+.f64 alpha i)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= unpow2_binary64 (pow.f64 beta 2))) (+.f64 alpha i)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (pow.f64 beta 2)) (Rewrite<= +-commutative_binary64 (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 i (/.f64 (pow.f64 beta 2) (+.f64 i alpha)))): 10 points increase in error, 9 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (+.f64 i alpha)) (pow.f64 beta 2))): 73 points increase in error, 5 points decrease in error

   5. Applied egg-rr 32.3

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

   6. Applied egg-rr 14.0

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.201056857348168 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + \frac{\beta}{\frac{i}{0.125}}\right) - \frac{\beta}{\frac{i}{0.125}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.201056857348168 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 2
Error	16.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7975981521892622 \cdot 10^{+226}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{\alpha}{\beta}}}\\ \end{array} \]

Alternative 3
Error	15.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.95290561208223 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\beta}\\ \end{array} \]

Alternative 4
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.80409000614123 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 5
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.80409000614123 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	16.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7975981521892622 \cdot 10^{+226}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	18.2
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022316 
(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)))