Octave 3.8, jcobi/2

Average Error: 24.6 → 1.4

Time: 24.4s

Precision: binary64

Cost: 28996

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

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 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999998)
     (/
      (+ (* 4.0 (/ i alpha)) (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (/
      (pow
       (pow
        (+
         1.0
         (*
          (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
          (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))))
        3.0)
       0.3333333333333333)
      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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999998) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else {
		tmp = pow(pow((1.0 + (((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta))))), 3.0), 0.3333333333333333) / 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(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999998)
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(((Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))))) ^ 3.0) ^ 0.3333333333333333) / 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[(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$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999998], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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)) < -0.999999980000000011

   1. Initial program 62.8

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

   2. Applied egg-rr 54.2

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]

   3. Taylor expanded in alpha around -inf 45.2

      \[\leadsto \frac{{\left({\color{blue}{\left(\left(\frac{{\beta}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{\alpha}\right) - \left(\frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right) \cdot \left(-1 \cdot \left(\beta + 2 \cdot i\right) + -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}} + -1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)}}^{3}\right)}^{0.3333333333333333}}{2} \]

   4. Simplified 45.2

      \[\leadsto \frac{{\left({\color{blue}{\left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} - \frac{\left(-\left(\beta + 2 \cdot i\right)\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha}\right) - \left(\frac{{\left(\left(-\left(\beta + 2 \cdot i\right)\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2 \cdot i}{\frac{\alpha \cdot \alpha}{\beta + \left(2 + 2 \cdot i\right)}}\right)\right)}}^{3}\right)}^{0.3333333333333333}}{2} \]
      Proof
      (-.f64 (-.f64 (*.f64 (/.f64 beta alpha) (/.f64 beta alpha)) (/.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 beta beta) (*.f64 alpha alpha))) (/.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 2 points increase in error, 4 points decrease in error
      (-.f64 (-.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 alpha alpha)) (/.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (/.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (*.f64 2 i)))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (neg.f64 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (-.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (-.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (/.f64 (-.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 beta (*.f64 -1 beta))) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha)) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (neg.f64 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha)))) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha)))) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 beta) beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (neg.f64 (+.f64 beta (*.f64 2 i))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (*.f64 2 i)))) (+.f64 beta (+.f64 2 (*.f64 2 i)))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (neg.f64 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 beta) beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (pow.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (Rewrite=> unpow2_binary64 (*.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 -1 beta))) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (-.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 -1 1) beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (-.f64 (*.f64 (Rewrite=> metadata-eval 0) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (-.f64 (Rewrite=> mul0-lft_binary64 0) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (Rewrite=> remove-double-neg_binary64 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (*.f64 alpha alpha) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (pow.f64 alpha 2)) (/.f64 (+.f64 beta (*.f64 2 i)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)) (+.f64 beta (+.f64 2 (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (-.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (pow.f64 alpha 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2))))): 3 points increase in error, 2 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (pow.f64 alpha 2)) (neg.f64 (/.f64 (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 2 (*.f64 2 i)))) (pow.f64 alpha 2)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (/.f64 (pow.f64 beta 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (-.f64 (+.f64 (*.f64 -1 beta) beta) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) alpha))) (+.f64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 -1 beta)) (*.f64 -1 (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i))))))) (+.f64 (*.f64 -1 (+.f64 beta (*.f64 2 i))) (*.f64 -1 (+.f64 beta (+.f64 2 (*.f64 2 i)))))) (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)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in alpha around inf 6.0

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

   6. Taylor expanded in i around 0 6.0

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

   if -0.999999980000000011 < (/.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 13.4

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

   2. Applied egg-rr 0.1

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	22468

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{1}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 3
Error	2.8
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.99999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4
Error	12.1
Cost	1612

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 6.981121166583162 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.041125755113839 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.060378385108493 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 5
Error	12.1
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 6.981121166583162 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 4.041125755113839 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.060378385108493 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.953228068292773 \cdot 10^{+181}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	17.6
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.0176859019342044 \cdot 10^{+22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	42.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(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))