Octave 3.8, jcobi/4

Average Error: 54.0 → 9.6

Time: 28.3s

Precision: binary64

Cost: 42948

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + i\right)\\ t_1 := {\left(\beta + \alpha\right)}^{2}\\ t_2 := -0.125 \cdot \frac{\beta + \alpha}{i}\\ t_3 := \left(\beta + \alpha\right) + 2 \cdot i\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625 + \left(\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - {\left(\frac{-1}{i}\right)}^{2} \cdot \left(\left(\beta \cdot \alpha + t_1\right) \cdot -0.0625 - \mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \left(\left(\beta + \alpha\right) \cdot -0.125 - 0.0625 \cdot \left(\left(\beta + \alpha\right) \cdot -2\right)\right), -0.00390625 \cdot \mathsf{fma}\left(4, -1 + t_1, t_1 \cdot 20\right)\right)\right)\right) + t_2\right)\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right) \cdot \frac{i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{t_0}}}{-1 + t_3 \cdot t_3}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;t_2 + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) + 1\right)}\\ \end{array} \]

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 (+ beta (+ alpha i)))
        (t_1 (pow (+ beta alpha) 2.0))
        (t_2 (* -0.125 (/ (+ beta alpha) i)))
        (t_3 (+ (+ beta alpha) (* 2.0 i))))
   (if (<= beta 1.5e+85)
     (+
      0.0625
      (+
       (-
        (* 0.0625 (/ (* 2.0 (+ beta alpha)) i))
        (*
         (pow (/ -1.0 i) 2.0)
         (-
          (* (+ (* beta alpha) t_1) -0.0625)
          (fma
           -2.0
           (*
            (+ beta alpha)
            (- (* (+ beta alpha) -0.125) (* 0.0625 (* (+ beta alpha) -2.0))))
           (* -0.00390625 (fma 4.0 (+ -1.0 t_1) (* t_1 20.0)))))))
       t_2))
     (if (<= beta 1.2e+127)
       (/
        (*
         (fma i t_0 (* beta alpha))
         (/ i (/ (pow (fma i 2.0 (+ beta alpha)) 2.0) t_0)))
        (+ -1.0 (* t_3 t_3)))
       (if (<= beta 1.5e+154)
         (+ t_2 (fma (* beta 0.125) (/ 1.0 i) 0.0625))
         (/
          (* i (/ (+ alpha i) (+ (fma i 2.0 alpha) (+ beta -1.0))))
          (fma i 2.0 (+ (+ beta alpha) 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 = beta + (alpha + i);
	double t_1 = pow((beta + alpha), 2.0);
	double t_2 = -0.125 * ((beta + alpha) / i);
	double t_3 = (beta + alpha) + (2.0 * i);
	double tmp;
	if (beta <= 1.5e+85) {
		tmp = 0.0625 + (((0.0625 * ((2.0 * (beta + alpha)) / i)) - (pow((-1.0 / i), 2.0) * ((((beta * alpha) + t_1) * -0.0625) - fma(-2.0, ((beta + alpha) * (((beta + alpha) * -0.125) - (0.0625 * ((beta + alpha) * -2.0)))), (-0.00390625 * fma(4.0, (-1.0 + t_1), (t_1 * 20.0))))))) + t_2);
	} else if (beta <= 1.2e+127) {
		tmp = (fma(i, t_0, (beta * alpha)) * (i / (pow(fma(i, 2.0, (beta + alpha)), 2.0) / t_0))) / (-1.0 + (t_3 * t_3));
	} else if (beta <= 1.5e+154) {
		tmp = t_2 + fma((beta * 0.125), (1.0 / i), 0.0625);
	} else {
		tmp = (i * ((alpha + i) / (fma(i, 2.0, alpha) + (beta + -1.0)))) / fma(i, 2.0, ((beta + alpha) + 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 = Float64(beta + Float64(alpha + i))
	t_1 = Float64(beta + alpha) ^ 2.0
	t_2 = Float64(-0.125 * Float64(Float64(beta + alpha) / i))
	t_3 = Float64(Float64(beta + alpha) + Float64(2.0 * i))
	tmp = 0.0
	if (beta <= 1.5e+85)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(Float64(2.0 * Float64(beta + alpha)) / i)) - Float64((Float64(-1.0 / i) ^ 2.0) * Float64(Float64(Float64(Float64(beta * alpha) + t_1) * -0.0625) - fma(-2.0, Float64(Float64(beta + alpha) * Float64(Float64(Float64(beta + alpha) * -0.125) - Float64(0.0625 * Float64(Float64(beta + alpha) * -2.0)))), Float64(-0.00390625 * fma(4.0, Float64(-1.0 + t_1), Float64(t_1 * 20.0))))))) + t_2));
	elseif (beta <= 1.2e+127)
		tmp = Float64(Float64(fma(i, t_0, Float64(beta * alpha)) * Float64(i / Float64((fma(i, 2.0, Float64(beta + alpha)) ^ 2.0) / t_0))) / Float64(-1.0 + Float64(t_3 * t_3)));
	elseif (beta <= 1.5e+154)
		tmp = Float64(t_2 + fma(Float64(beta * 0.125), Float64(1.0 / i), 0.0625));
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / Float64(fma(i, 2.0, alpha) + Float64(beta + -1.0)))) / fma(i, 2.0, Float64(Float64(beta + alpha) + 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[(beta + N[(alpha + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(-0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+85], N[(0.0625 + N[(N[(N[(0.0625 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(-1.0 / i), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision] * -0.0625), $MachinePrecision] - N[(-2.0 * N[(N[(beta + alpha), $MachinePrecision] * N[(N[(N[(beta + alpha), $MachinePrecision] * -0.125), $MachinePrecision] - N[(0.0625 * N[(N[(beta + alpha), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.00390625 * N[(4.0 * N[(-1.0 + t$95$1), $MachinePrecision] + N[(t$95$1 * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.2e+127], N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[Power[N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+154], N[(t$95$2 + N[(N[(beta * 0.125), $MachinePrecision] * N[(1.0 / i), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(N[(i * 2.0 + alpha), $MachinePrecision] + N[(beta + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * 2.0 + N[(N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.5e85

   1. Initial program 47.9

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

   2. Simplified 42.4

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \cdot \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \]
      Proof
      (*.f64 (/.f64 i (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1)) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))) (/.f64 (*.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha)))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha)))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.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 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (*.f64 (+.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)))))): 30 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_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, 8 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, 12 points decrease in error

   3. Taylor expanded in i around -inf 2.1

      \[\leadsto \color{blue}{\left(0.0625 + \left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} + -1 \cdot \left({\left(\frac{-1}{i}\right)}^{2} \cdot \left(0.0625 \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) - \left(-2 \cdot \left(\left(0.0625 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.125 \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) + -0.00390625 \cdot \left(16 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + 4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right)\right)\right)\right)\right)\right)\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   4. Simplified 2.1

      \[\leadsto \color{blue}{0.0625 + \left(\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - {\left(\frac{-1}{i}\right)}^{2} \cdot \left(0.0625 \cdot \left(\left(-\beta \cdot \alpha\right) - {\left(\beta + \alpha\right)}^{2}\right) - \mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \left(\left(2 \cdot \left(\beta + \alpha\right)\right) \cdot 0.0625 + -0.125 \cdot \left(\beta + \alpha\right)\right), -0.00390625 \cdot \mathsf{fma}\left(4, {\left(\beta + \alpha\right)}^{2} + -1, {\left(\beta + \alpha\right)}^{2} \cdot 20\right)\right)\right)\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\right)} \]
      Proof
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (*.f64 2 (+.f64 beta alpha)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (-.f64 (neg.f64 (*.f64 beta alpha)) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (-.f64 (neg.f64 (*.f64 beta alpha)) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta))) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (-.f64 (neg.f64 (*.f64 beta alpha)) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (neg.f64 (pow.f64 (+.f64 beta alpha) 2))))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2))))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (*.f64 2 (+.f64 beta alpha)) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) 1/16) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha)))) (*.f64 -1/8 (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (+.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/8)) (+.f64 beta alpha)))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (+.f64 beta alpha) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) -1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (fma.f64 4 (+.f64 (pow.f64 (+.f64 beta alpha) 2) (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (fma.f64 4 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 (pow.f64 (+.f64 beta alpha) 2) 20))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (fma.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1) (*.f64 (pow.f64 (+.f64 beta alpha) 2) (Rewrite<= metadata-eval (+.f64 16 4))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (fma.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))) (*.f64 4 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))) (*.f64 16 (pow.f64 (+.f64 beta alpha) 2))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 8 points increase in error, 4 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (fma.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha)) (*.f64 -1/256 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (-.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (neg.f64 (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (+.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))))))) (*.f64 -1/8 (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (+.f64 (+.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 -1 (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)))))))))) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/8)) (/.f64 (+.f64 beta alpha) i)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 -1 (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1)))))))))) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1/16 (+.f64 (*.f64 1/16 (/.f64 (+.f64 (*.f64 2 alpha) (*.f64 2 beta)) i)) (*.f64 -1 (*.f64 (pow.f64 (/.f64 -1 i) 2) (-.f64 (*.f64 1/16 (+.f64 (*.f64 -1 (*.f64 beta alpha)) (*.f64 -1 (pow.f64 (+.f64 beta alpha) 2)))) (+.f64 (*.f64 -2 (*.f64 (-.f64 (*.f64 1/16 (+.f64 (*.f64 2 beta) (*.f64 2 alpha))) (*.f64 1/8 (+.f64 beta alpha))) (+.f64 beta alpha))) (*.f64 -1/256 (+.f64 (*.f64 16 (pow.f64 (+.f64 beta alpha) 2)) (+.f64 (*.f64 4 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 4 (-.f64 (pow.f64 (+.f64 beta alpha) 2) 1))))))))))) (*.f64 1/8 (/.f64 (+.f64 beta alpha) i)))): 1 points increase in error, 0 points decrease in error

   if 1.5e85 < beta < 1.2000000000000001e127

   1. Initial program 55.4

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

   2. Applied egg-rr 55.4

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

   3. Simplified 32.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i, \left(i + \alpha\right) + \beta, \beta \cdot \alpha\right) \cdot \frac{i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\left(i + \alpha\right) + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (*.f64 (fma.f64 i (+.f64 (+.f64 i alpha) beta) (*.f64 beta alpha)) (/.f64 i (/.f64 (pow.f64 (fma.f64 i 2 (+.f64 beta alpha)) 2) (+.f64 (+.f64 i alpha) beta)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (fma.f64 i (Rewrite<= associate-+r+_binary64 (+.f64 i (+.f64 alpha beta))) (*.f64 beta alpha)) (/.f64 i (/.f64 (pow.f64 (fma.f64 i 2 (+.f64 beta alpha)) 2) (+.f64 (+.f64 i alpha) beta)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (Rewrite=> *-commutative_binary64 (*.f64 alpha beta))) (/.f64 i (/.f64 (pow.f64 (fma.f64 i 2 (+.f64 beta alpha)) 2) (+.f64 (+.f64 i alpha) beta)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 i (/.f64 (pow.f64 (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))) 2) (+.f64 (+.f64 i alpha) beta)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 i (/.f64 (pow.f64 (fma.f64 i 2 (+.f64 alpha beta)) 2) (Rewrite<= associate-+r+_binary64 (+.f64 i (+.f64 alpha beta)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (pow.f64 (fma.f64 i 2 (+.f64 alpha beta)) 2)))): 8 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (*.f64 i (+.f64 i (+.f64 alpha beta)))) (pow.f64 (fma.f64 i 2 (+.f64 alpha beta)) 2))): 61 points increase in error, 4 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))) (pow.f64 (fma.f64 i 2 (+.f64 alpha beta)) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))) (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))))) (pow.f64 (fma.f64 i 2 (+.f64 alpha beta)) 2)): 0 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))) (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta))))) (Rewrite=> unpow2_binary64 (*.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))) (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (sqrt.f64 (*.f64 (*.f64 i (+.f64 i (+.f64 alpha beta))) (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)))) (fma.f64 i 2 (+.f64 alpha beta))))): 5 points increase in error, 6 points decrease in error

   if 1.2000000000000001e127 < beta < 1.50000000000000013e154

   1. Initial program 61.0

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

   2. Simplified 31.7

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right), \beta + \mathsf{fma}\left(i, 2, \alpha\right), -1\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\frac{\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}{i + \left(\alpha + \beta\right)}}} \]
      Proof
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha)))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i))))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.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))) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 48 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.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))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (*.f64 (+.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)))))): 30 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_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, 8 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, 12 points decrease in error

   3. Taylor expanded in i around inf 28.1

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

   4. Taylor expanded in alpha around 0 28.1

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

   5. Applied egg-rr 28.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125 \cdot \beta, \frac{1}{i}, 0.0625\right)} - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   if 1.50000000000000013e154 < 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 49.9

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

   3. Applied egg-rr 33.9

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

   4. Applied egg-rr 31.8

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

   5. Simplified 15.5

      \[\leadsto \color{blue}{\frac{i \cdot \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, 1 + \left(\alpha + \beta\right)\right)}} \]
      Proof
      (/.f64 (*.f64 i (/.f64 (+.f64 i alpha) (+.f64 (fma.f64 i 2 alpha) (+.f64 beta -1)))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 i alpha) 1)) (+.f64 (fma.f64 i 2 alpha) (+.f64 beta -1)))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 (fma.f64 i 2 alpha) (+.f64 beta (Rewrite<= metadata-eval (neg.f64 1)))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 (fma.f64 i 2 alpha) (Rewrite<= sub-neg_binary64 (-.f64 beta 1))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (fma.f64 i 2 alpha) beta) 1)))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 beta (fma.f64 i 2 alpha))) 1))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (-.f64 (+.f64 beta (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 i 2) alpha))) 1))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (-.f64 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 alpha (*.f64 i 2)))) 1))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (-.f64 (+.f64 beta (+.f64 alpha (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) 1))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (Rewrite=> associate--l+_binary64 (+.f64 beta (-.f64 (+.f64 alpha (*.f64 2 i)) 1))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 beta (Rewrite=> associate--l+_binary64 (+.f64 alpha (-.f64 (*.f64 2 i) 1)))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 beta (+.f64 alpha (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 i 2)) 1))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 beta (+.f64 alpha (Rewrite=> fma-neg_binary64 (fma.f64 i 2 (neg.f64 1))))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 beta (+.f64 alpha (fma.f64 i 2 (Rewrite=> metadata-eval -1)))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 i (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 i alpha) (/.f64 1 (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1))))))) (fma.f64 i 2 (+.f64 1 (+.f64 alpha beta)))): 11 points increase in error, 24 points decrease in error
      (/.f64 (*.f64 i (*.f64 (+.f64 i alpha) (/.f64 1 (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1)))))) (fma.f64 i 2 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1)) (*.f64 (+.f64 i alpha) (/.f64 1 (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1)))))))): 28 points increase in error, 12 points decrease in error
      (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 i alpha) 1) (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1))))))): 26 points increase in error, 6 points decrease in error
      (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1)) (/.f64 (Rewrite=> *-rgt-identity_binary64 (+.f64 i alpha)) (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 i (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1)) (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1)))) (+.f64 i alpha)))): 143 points increase in error, 3 points decrease in error
      (/.f64 i (Rewrite<= associate-*r/_binary64 (*.f64 (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1)) (/.f64 (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1))) (+.f64 i alpha))))): 9 points increase in error, 140 points decrease in error
      (/.f64 i (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 beta (+.f64 alpha (fma.f64 i 2 -1))) (+.f64 i alpha)) (fma.f64 i 2 (+.f64 (+.f64 alpha beta) 1))))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625 + \left(\left(0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i} - {\left(\frac{-1}{i}\right)}^{2} \cdot \left(\left(\beta \cdot \alpha + {\left(\beta + \alpha\right)}^{2}\right) \cdot -0.0625 - \mathsf{fma}\left(-2, \left(\beta + \alpha\right) \cdot \left(\left(\beta + \alpha\right) \cdot -0.125 - 0.0625 \cdot \left(\left(\beta + \alpha\right) \cdot -2\right)\right), -0.00390625 \cdot \mathsf{fma}\left(4, -1 + {\left(\beta + \alpha\right)}^{2}, {\left(\beta + \alpha\right)}^{2} \cdot 20\right)\right)\right)\right) + -0.125 \cdot \frac{\beta + \alpha}{i}\right)\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, \beta + \left(\alpha + i\right), \beta \cdot \alpha\right) \cdot \frac{i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\beta + \left(\alpha + i\right)}}}{-1 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	22152

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := \beta + \left(\alpha + i\right)\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right) \cdot \frac{i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{t_1}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) + 1\right)}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	21192

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.55 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(\beta + i\right) \cdot \frac{1}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{\beta + i}}\right) \cdot \frac{i \cdot i}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}}\\ \mathbf{elif}\;\beta \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) + 1\right)}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	14540

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+127}:\\ \;\;\;\;\frac{i \cdot i}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{\left(\beta + i\right) \cdot \left(\beta + i\right)}{\beta \cdot \beta + \left(\beta + i\right) \cdot \left(i \cdot 4\right)}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+155}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)} \cdot \frac{\alpha + i}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, -1\right)\right)}\\ \end{array} \]

Alternative 4
Error	9.8
Cost	14540

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{i \cdot i}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{\left(\beta + i\right) \cdot \left(\beta + i\right)}{\beta \cdot \beta + \left(\beta + i\right) \cdot \left(i \cdot 4\right)}\\ \mathbf{elif}\;\beta \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + -1\right)}}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) + 1\right)}\\ \end{array} \]

Alternative 5
Error	10.5
Cost	8712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{i \cdot i}{-1 + {\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{\left(\beta + i\right) \cdot \left(\beta + i\right)}{\beta \cdot \beta + \left(\beta + i\right) \cdot \left(i \cdot 4\right)}\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\alpha + i}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 6
Error	11.1
Cost	7884

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta + \left(-1 + 2 \cdot i\right)}}{\beta + \left(1 + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 1.16 \cdot 10^{+154}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\left(\alpha + i\right) \cdot \frac{i}{\beta}\right)\\ \end{array} \]

Alternative 7
Error	11.2
Cost	7884

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta + \left(-1 + 2 \cdot i\right)}}{\beta + \left(1 + 2 \cdot i\right)}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+153}:\\ \;\;\;\;-0.125 \cdot \frac{\beta + \alpha}{i} + \mathsf{fma}\left(\beta \cdot 0.125, \frac{1}{i}, 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\alpha + i}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Error	12.4
Cost	6852

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(\left(\beta + \alpha\right) + i\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(\beta \cdot \alpha + t_2\right)}{t_1}}{-1 + t_1}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 9
Error	17.1
Cost	1612

\[\begin{array}{l} t_0 := \frac{\frac{i \cdot i}{\beta + \left(-1 + 2 \cdot i\right)}}{\beta + \left(1 + 2 \cdot i\right)}\\ \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 7.5 \cdot 10^{+155}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	16.0
Cost	1352

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{-1 + \beta \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 11
Error	16.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+127}:\\ \;\;\;\;\frac{i \cdot i}{-1 + \beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta \cdot -0.125}{i}\\ \end{array} \]

Alternative 12
Error	18.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+124}:\\ \;\;\;\;\left(i \cdot i\right) \cdot \frac{\frac{1}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.65 \cdot 10^{+255}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	18.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10^{+86}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{i \cdot i}{-1 + \beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+255}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	16.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75 \cdot 10^{+255}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Error	57.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022325 
(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)))