Octave 3.8, jcobi/2

Average Error: 23.4 → 1.5

Time: 21.6s

Precision: binary64

Cost: 55876

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(t_0 + \left(\frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} - t_0 \cdot t_0\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}\\ \end{array} \]

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 (/ (fma i 4.0 (fma beta 2.0 2.0)) alpha))
        (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.5)
     (/
      (+
       (* (/ beta alpha) (/ beta alpha))
       (+
        t_0
        (-
         (* (/ (+ beta (fma 2.0 i 2.0)) alpha) (/ (fma 2.0 i beta) alpha))
         (* t_0 t_0))))
      2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
        (+ alpha (fma 2.0 i (+ beta 2.0))))
       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 = fma(i, 4.0, fma(beta, 2.0, 2.0)) / alpha;
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = (((beta / alpha) * (beta / alpha)) + (t_0 + ((((beta + fma(2.0, i, 2.0)) / alpha) * (fma(2.0, i, beta) / alpha)) - (t_0 * t_0)))) / 2.0;
	} else {
		tmp = fma((alpha + beta), (((beta - alpha) / (alpha + fma(2.0, i, beta))) / (alpha + fma(2.0, i, (beta + 2.0)))), 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(fma(i, 4.0, fma(beta, 2.0, 2.0)) / alpha)
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta / alpha) * Float64(beta / alpha)) + Float64(t_0 + Float64(Float64(Float64(Float64(beta + fma(2.0, i, 2.0)) / alpha) * Float64(fma(2.0, i, beta) / alpha)) - Float64(t_0 * t_0)))) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))) / Float64(alpha + fma(2.0, i, Float64(beta + 2.0)))), 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[(N[(i * 4.0 + N[(beta * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[(N[(N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + N[(beta + 2.0), $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)) < -0.5

   1. Initial program 61.4

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

   2. Applied egg-rr 53.0

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

   3. Taylor expanded in alpha around inf 14.0

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

   4. Simplified 6.3

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\left(\frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha} - \frac{\beta + \mathsf{fma}\left(2, i, 2\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha}\right) - \frac{\mathsf{fma}\left(i, 4, \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\right)}}{2} \]
      Proof
      (-.f64 (+.f64 0 (*.f64 (/.f64 beta alpha) (/.f64 beta alpha))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 beta alpha))) (*.f64 (/.f64 beta alpha) (/.f64 beta alpha))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 beta alpha)) (*.f64 (/.f64 beta alpha) (/.f64 beta alpha))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 beta alpha) (*.f64 -1 (/.f64 beta alpha)))) (*.f64 (/.f64 beta alpha) (/.f64 beta alpha))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (/.f64 beta alpha) (*.f64 -1 (/.f64 beta alpha))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 beta beta) (*.f64 alpha alpha)))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 10 points increase in error, 2 points decrease in error
      (-.f64 (+.f64 (+.f64 (/.f64 beta alpha) (*.f64 -1 (/.f64 beta alpha))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 alpha alpha))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (+.f64 (/.f64 beta alpha) (*.f64 -1 (/.f64 beta alpha))) (/.f64 (pow.f64 beta 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2))))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 beta 2) 2))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 beta)) 2)) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (fma.f64 i 4 (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 beta)))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 4) (+.f64 2 (*.f64 2 beta)))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 i)) (+.f64 2 (*.f64 2 beta))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) beta) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 beta (*.f64 -1 beta))) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 beta 2) 2))) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 beta)) 2)) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (fma.f64 i 4 (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 beta)))) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 4) (+.f64 2 (*.f64 2 beta)))) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (*.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) alpha) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 i)) (+.f64 2 (*.f64 2 beta))) alpha)) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (*.f64 alpha alpha))) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 13 points increase in error, 16 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta (fma.f64 2 i 2)) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2))) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (+.f64 beta (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 i)))) alpha) (/.f64 (fma.f64 2 i beta) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (+.f64 beta (+.f64 2 (*.f64 2 i))) alpha) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (*.f64 (/.f64 (+.f64 beta (+.f64 2 (*.f64 2 i))) alpha) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta (+.f64 2 (*.f64 2 i))) (+.f64 beta (*.f64 2 i))) (*.f64 alpha alpha)))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 1 points increase in error, 1 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i))))) (*.f64 alpha alpha))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (neg.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)))) (/.f64 (fma.f64 i 4 (fma.f64 beta 2 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (/.f64 (fma.f64 i 4 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 beta 2) 2))) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (/.f64 (fma.f64 i 4 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 beta)) 2)) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (/.f64 (fma.f64 i 4 (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 2 beta)))) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 4) (+.f64 2 (*.f64 2 beta)))) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 i)) (+.f64 2 (*.f64 2 beta))) alpha))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (neg.f64 (/.f64 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))) alpha))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))) alpha))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 beta alpha) (+.f64 (*.f64 -1 (/.f64 beta alpha)) (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))) alpha)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))) (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta))))) (+.f64 (*.f64 4 i) (+.f64 2 (*.f64 2 beta)))) (pow.f64 alpha 2)))))): 0 points increase in error, 0 points decrease in error

   if -0.5 < (/.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 12.1

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

   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): 66 points increase in error, 18 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, 14 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): 14 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, 14 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): 4 points increase in error, 2 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.4
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	3268

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	14.4
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.51498093579719 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 5.216822500388089 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.42245857262143 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.3
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.51498093579719 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 5.216822500388089 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.42245857262143 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	972

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.51498093579719 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.216822500388089 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.42245857262143 \cdot 10^{+118}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	7.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.51498093579719 \cdot 10^{+27}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	16.1
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 2.0151862494336248 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.52385673549468 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.706767660536565 \cdot 10^{+262}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	19.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.9300257646631184 \cdot 10^{-291}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 4.5002291920124785 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 5.824886419559466 \cdot 10^{+40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	17.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.824886419559466 \cdot 10^{+40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	24.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022306 
(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))