Octave 3.8, jcobi/2

Average Error: 23.9 → 1.3

Time: 23.6s

Precision: binary64

Cost: 76356

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \mathsf{fma}\left(2, i, \beta\right)\\ t_1 := \beta + \mathsf{fma}\left(2, i, 2\right)\\ t_2 := \mathsf{fma}\left(\beta, -1, t_1\right)\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_3}}{2 + t_3} \leq -0.98:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, i, \beta\right) + \left(\beta + t_2\right)}{\alpha} - \left(\left(\frac{t_2}{\alpha} \cdot \frac{t_1}{\alpha} + \frac{t_2}{\alpha \cdot \alpha} \cdot t_0\right) + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{t_0}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{1}{\left(\alpha + t_1\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (fma 2.0 i beta)))
        (t_1 (+ beta (fma 2.0 i 2.0)))
        (t_2 (fma beta -1.0 t_1))
        (t_3 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_3) (+ 2.0 t_3)) -0.98)
     (/
      (-
       (/ (+ (fma 2.0 i beta) (+ beta t_2)) alpha)
       (+
        (+ (* (/ t_2 alpha) (/ t_1 alpha)) (* (/ t_2 (* alpha alpha)) t_0))
        (* (/ (fma 2.0 i beta) alpha) (/ t_0 alpha))))
      2.0)
     (/
      (fma
       (+ alpha beta)
       (/ 1.0 (* (+ alpha t_1) (/ (fma 2.0 i (+ alpha beta)) (- beta alpha))))
       1.0)
      2.0))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = beta + fma(2.0, i, beta);
	double t_1 = beta + fma(2.0, i, 2.0);
	double t_2 = fma(beta, -1.0, t_1);
	double t_3 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_3) / (2.0 + t_3)) <= -0.98) {
		tmp = (((fma(2.0, i, beta) + (beta + t_2)) / alpha) - ((((t_2 / alpha) * (t_1 / alpha)) + ((t_2 / (alpha * alpha)) * t_0)) + ((fma(2.0, i, beta) / alpha) * (t_0 / alpha)))) / 2.0;
	} else {
		tmp = fma((alpha + beta), (1.0 / ((alpha + t_1) * (fma(2.0, i, (alpha + beta)) / (beta - alpha)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta, i)
	t_0 = Float64(beta + fma(2.0, i, beta))
	t_1 = Float64(beta + fma(2.0, i, 2.0))
	t_2 = fma(beta, -1.0, t_1)
	t_3 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_3) / Float64(2.0 + t_3)) <= -0.98)
		tmp = Float64(Float64(Float64(Float64(fma(2.0, i, beta) + Float64(beta + t_2)) / alpha) - Float64(Float64(Float64(Float64(t_2 / alpha) * Float64(t_1 / alpha)) + Float64(Float64(t_2 / Float64(alpha * alpha)) * t_0)) + Float64(Float64(fma(2.0, i, beta) / alpha) * Float64(t_0 / alpha)))) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(1.0 / Float64(Float64(alpha + t_1) * Float64(fma(2.0, i, Float64(alpha + beta)) / Float64(beta - alpha)))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta * -1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(N[(N[(2.0 * i + beta), $MachinePrecision] + N[(beta + t$95$2), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(N[(t$95$2 / alpha), $MachinePrecision] * N[(t$95$1 / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(1.0 / N[(N[(alpha + t$95$1), $MachinePrecision] * N[(N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.97999999999999998

   1. Initial program 61.7

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

   2. Simplified 53.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \beta\right) + \left(\alpha + 2\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}} \]
      Proof
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (fma.f64 2 i beta) (+.f64 alpha 2))) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)) (+.f64 alpha 2))) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) (+.f64 alpha 2))) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha) 2))) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) 2)) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 2)) (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) 1)) 2): 1 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 83 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in alpha around -inf 13.5

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

   4. Simplified 5.7

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{\mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha \cdot \alpha} \cdot \left(\left(-\beta\right) - \mathsf{fma}\left(2, i, \beta\right)\right) - \frac{\mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha}\right) + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\left(-\beta\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right) - \frac{\left(\left(-\beta\right) - \mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) - \mathsf{fma}\left(2, i, \beta\right)}{\alpha}}}{2} \]
      Proof
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2)))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i))))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (fma.f64 beta -1 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (fma.f64 beta -1 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 beta -1) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 beta)) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (*.f64 alpha alpha)) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (-.f64 (neg.f64 beta) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (pow.f64 alpha 2)) (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)) (fma.f64 2 i beta))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 i) beta)))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (/.f64 (pow.f64 alpha 2) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 1 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 13 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2)))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (fma.f64 beta -1 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i))))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (fma.f64 beta -1 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (fma.f64 beta -1 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 beta -1) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 beta)) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) alpha) (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) alpha) (/.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2))) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) alpha) (/.f64 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i)))) alpha))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (*.f64 alpha alpha)))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 11 points increase in error, 11 points decrease in error
      (-.f64 (+.f64 (-.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (neg.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (*.f64 (/.f64 (fma.f64 2 i beta) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 i) beta)) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) alpha) (/.f64 (-.f64 (neg.f64 beta) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta (*.f64 2 i)) alpha) (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)) (fma.f64 2 i beta)) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta (*.f64 2 i)) alpha) (/.f64 (-.f64 (*.f64 -1 beta) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 i) beta))) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta (*.f64 2 i)) alpha) (/.f64 (-.f64 (*.f64 -1 beta) (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) alpha))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (*.f64 alpha alpha)))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 19 points increase in error, 4 points decrease in error
      (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2))))) (/.f64 (-.f64 (-.f64 (neg.f64 beta) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)) (fma.f64 beta -1 (+.f64 beta (fma.f64 2 i 2)))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (fma.f64 beta -1 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (fma.f64 beta -1 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i)))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (fma.f64 beta -1 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (fma.f64 beta -1 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 beta -1) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (-.f64 (*.f64 -1 beta) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 beta)) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 beta) (neg.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (fma.f64 2 i beta)) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 i) beta))) alpha)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) alpha)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (neg.f64 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 beta (*.f64 2 i))) alpha)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 beta (*.f64 2 i))) alpha)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 beta (*.f64 2 i))) alpha)) (+.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))) (-.f64 (*.f64 -1 beta) (+.f64 beta (*.f64 2 i)))) (pow.f64 alpha 2)))))): 0 points increase in error, 0 points decrease in error

   if -0.97999999999999998 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 12.7

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}} \]
      Proof
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) (+.f64 beta 2))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 i) beta) 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 63 points increase in error, 21 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= metadata-eval (neg.f64 -1))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) -1)) 2): 8 points increase in error, 6 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta))) -1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> fma-neg_binary64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (neg.f64 -1))) 2): 6 points increase in error, 8 points decrease in error
      (/.f64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (Rewrite=> metadata-eval 1)) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta)) 1)) 2): 8 points increase in error, 6 points decrease in error
      (/.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 87 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \color{blue}{\frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}}, 1\right)}{2} \]
      Proof
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 beta (fma.f64 2 i 2))) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 beta (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 i) 2)))) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i))))) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 2 (*.f64 2 i)) beta))) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 i) 2)) beta)) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 (Rewrite<= fma-udef_binary64 (fma.f64 2 i 2)) beta)) (/.f64 (fma.f64 2 i (+.f64 beta alpha)) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 (+.f64 alpha (+.f64 (fma.f64 2 i 2) beta)) (/.f64 (fma.f64 2 i (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))) (-.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow-1_binary64 (pow.f64 (*.f64 (+.f64 alpha (+.f64 (fma.f64 2 i 2) beta)) (/.f64 (fma.f64 2 i (+.f64 alpha beta)) (-.f64 beta alpha))) -1)): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.98:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(2, i, \beta\right) + \left(\beta + \mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}{\alpha} - \left(\left(\frac{\mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha} + \frac{\mathsf{fma}\left(\beta, -1, \beta + \mathsf{fma}\left(2, i, 2\right)\right)}{\alpha \cdot \alpha} \cdot \left(\beta + \mathsf{fma}\left(2, i, \beta\right)\right)\right) + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\beta + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	55876

\[\begin{array}{l} t_0 := \beta + \mathsf{fma}\left(2, i, 2\right)\\ t_1 := \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{2 + t_2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(t_1 + \left(\frac{\mathsf{fma}\left(2, i, \beta\right)}{\frac{\alpha \cdot \alpha}{t_0}} - t_1 \cdot t_1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{1}{\left(\alpha + t_0\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	22468

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{1}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}, 1\right)}{2}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}}{2}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	3524

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.98:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 5
Error	2.2
Cost	2628

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 6
Error	9.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	13.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	15.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	12.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	14.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 12
Error	17.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	24.5
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022328 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))