Octave 3.8, jcobi/4

Average Error: 54.3 → 9.7

Time: 18.6s

Precision: binary64

Cost: 14796

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 3.0607266517229465 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i} \cdot \frac{\beta}{i}, 0.0625, 0.0625\right) + \frac{\mathsf{fma}\left(\beta, \beta, -1\right) \cdot -0.015625 + \left(\beta \cdot \beta\right) \cdot -0.078125}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.200716879273244e+102)
   0.0625
   (if (<= beta 1.4270341292843418e+110)
     (/ i (/ (* beta beta) i))
     (if (<= beta 3.0607266517229465e+134)
       (+
        (fma (* (/ beta i) (/ beta i)) 0.0625 0.0625)
        (/
         (+ (* (fma beta beta -1.0) -0.015625) (* (* beta beta) -0.078125))
         (* i i)))
       (* (/ (+ i alpha) beta) (/ i beta))))))

C

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

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.200716879273244e+102) {
		tmp = 0.0625;
	} else if (beta <= 1.4270341292843418e+110) {
		tmp = i / ((beta * beta) / i);
	} else if (beta <= 3.0607266517229465e+134) {
		tmp = fma(((beta / i) * (beta / i)), 0.0625, 0.0625) + (((fma(beta, beta, -1.0) * -0.015625) + ((beta * beta) * -0.078125)) / (i * i));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.200716879273244e+102)
		tmp = 0.0625;
	elseif (beta <= 1.4270341292843418e+110)
		tmp = Float64(i / Float64(Float64(beta * beta) / i));
	elseif (beta <= 3.0607266517229465e+134)
		tmp = Float64(fma(Float64(Float64(beta / i) * Float64(beta / i)), 0.0625, 0.0625) + Float64(Float64(Float64(fma(beta, beta, -1.0) * -0.015625) + Float64(Float64(beta * beta) * -0.078125)) / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.200716879273244e+102], 0.0625, If[LessEqual[beta, 1.4270341292843418e+110], N[(i / N[(N[(beta * beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.0607266517229465e+134], N[(N[(N[(N[(beta / i), $MachinePrecision] * N[(beta / i), $MachinePrecision]), $MachinePrecision] * 0.0625 + 0.0625), $MachinePrecision] + N[(N[(N[(N[(beta * beta + -1.0), $MachinePrecision] * -0.015625), $MachinePrecision] + N[(N[(beta * beta), $MachinePrecision] * -0.078125), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.20071687927324382e102

   1. Initial program 48.8

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

   2. Taylor expanded in i around inf 3.5

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

   3. Taylor expanded in beta around 0 3.4

      \[\leadsto \color{blue}{0.0625} \]

   if 4.20071687927324382e102 < beta < 1.42703412928434182e110

   1. Initial program 52.1

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

   2. Simplified 26.3

      \[\leadsto \color{blue}{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 i (+.f64 beta (+.f64 i alpha))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta i) alpha))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta i)))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) i))) (/.f64 (/.f64 (fma.f64 i (+.f64 beta (+.f64 i alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta i) alpha)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta i))) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (fma.f64 i 2 beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (fma.f64 (+.f64 alpha (fma.f64 i 2 beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (fma.f64 i 2 beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i)))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))))): 61 points increase in error, 3 points decrease in error
      (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (*.f64 i (+.f64 (+.f64 alpha beta) i)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 74 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))): 3 points increase in error, 3 points decrease in error

   3. Taylor expanded in beta around inf 36.7

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

   4. Simplified 36.0

      \[\leadsto \color{blue}{\frac{i}{\beta \cdot \beta} \cdot \left(\alpha + i\right)} \]
      Proof
      (*.f64 (/.f64 i (*.f64 beta beta)) (+.f64 alpha i)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= unpow2_binary64 (pow.f64 beta 2))) (+.f64 alpha i)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (pow.f64 beta 2)) (Rewrite<= +-commutative_binary64 (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 i (/.f64 (pow.f64 beta 2) (+.f64 i alpha)))): 6 points increase in error, 13 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (+.f64 i alpha)) (pow.f64 beta 2))): 61 points increase in error, 7 points decrease in error

   5. Taylor expanded in i around inf 38.4

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]

   6. Simplified 37.7

      \[\leadsto \color{blue}{i \cdot \frac{i}{\beta \cdot \beta}} \]
      Proof
      (*.f64 i (/.f64 i (*.f64 beta beta))): 0 points increase in error, 0 points decrease in error
      (*.f64 i (/.f64 i (Rewrite<= unpow2_binary64 (pow.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 i i) (pow.f64 beta 2))): 57 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) (pow.f64 beta 2)): 1 points increase in error, 0 points decrease in error

   7. Applied egg-rr 37.7

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{i}}} \]

   if 1.42703412928434182e110 < beta < 3.06072665172294649e134

   1. Initial program 59.0

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

   2. Taylor expanded in alpha around 0 63.7

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

   3. Simplified 35.3

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\beta + i \cdot 2\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + i \cdot 2\right)}^{2} + -1}} \]
      Proof
      (*.f64 (/.f64 (*.f64 i i) (pow.f64 (+.f64 beta (*.f64 i 2)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 i 2)) 2) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) (pow.f64 (+.f64 beta (*.f64 i 2)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 i 2)) 2) -1))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 i 2)) 2) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))) 2) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta i) 2)) (*.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) (-.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) 1)))): 59 points increase in error, 0 points decrease in error

   4. Taylor expanded in i around inf 24.3

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

   5. Simplified 24.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{i} \cdot \frac{\beta}{i}, 0.0625, 0.0625\right) - \frac{\mathsf{fma}\left(\beta, \beta, -1\right) \cdot 0.015625 + \left(\beta \cdot \beta\right) \cdot 0.078125}{i \cdot i}} \]
      Proof
      (-.f64 (fma.f64 (*.f64 (/.f64 beta i) (/.f64 beta i)) 1/16 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 beta beta) (*.f64 i i))) 1/16 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 4 points increase in error, 5 points decrease in error
      (-.f64 (fma.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (*.f64 i i)) 1/16 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 (/.f64 (pow.f64 beta 2) (Rewrite<= unpow2_binary64 (pow.f64 i 2))) 1/16 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 1 points decrease in error
      (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (pow.f64 beta 2) (pow.f64 i 2)) 1/16) 1/16)) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2)))) 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta -1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (fma.f64 beta beta (Rewrite<= metadata-eval (neg.f64 1))) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 beta beta) 1)) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) 1) 1/64) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (-.f64 (pow.f64 beta 2) 1) (Rewrite<= metadata-eval (*.f64 4 1/256))) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 beta 2) 1) 4) 1/256)) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1))) 1/256) (*.f64 (*.f64 beta beta) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) 1/256) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) 5/64)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) 1/256) (*.f64 (pow.f64 beta 2) (Rewrite<= metadata-eval (*.f64 20 1/256)))) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) 1/256) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 beta 2) 20) 1/256))) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) 1/256) (*.f64 (*.f64 (pow.f64 beta 2) (Rewrite<= metadata-eval (+.f64 4 16))) 1/256)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (+.f64 (*.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) 1/256) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 4 (pow.f64 beta 2)) (*.f64 16 (pow.f64 beta 2)))) 1/256)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/256 (+.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) (+.f64 (*.f64 4 (pow.f64 beta 2)) (*.f64 16 (pow.f64 beta 2)))))) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (/.f64 (*.f64 1/256 (+.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) (+.f64 (*.f64 4 (pow.f64 beta 2)) (*.f64 16 (pow.f64 beta 2))))) (Rewrite<= unpow2_binary64 (pow.f64 i 2)))): 1 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 1/16 (/.f64 (pow.f64 beta 2) (pow.f64 i 2))) 1/16) (Rewrite<= associate-*r/_binary64 (*.f64 1/256 (/.f64 (+.f64 (*.f64 4 (-.f64 (pow.f64 beta 2) 1)) (+.f64 (*.f64 4 (pow.f64 beta 2)) (*.f64 16 (pow.f64 beta 2)))) (pow.f64 i 2))))): 0 points increase in error, 0 points decrease in error

   if 3.06072665172294649e134 < beta

   1. Initial program 63.8

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

   2. Taylor expanded in beta around inf 47.5

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

   3. Simplified 18.2

      \[\leadsto \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]
      Proof
      (*.f64 (/.f64 (+.f64 i alpha) beta) (/.f64 i beta)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 i alpha) i) (*.f64 beta beta))): 86 points increase in error, 12 points decrease in error
      (/.f64 (*.f64 (+.f64 i alpha) i) (Rewrite<= unpow2_binary64 (pow.f64 beta 2))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 3.0607266517229465 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i} \cdot \frac{\beta}{i}, 0.0625, 0.0625\right) + \frac{\mathsf{fma}\left(\beta, \beta, -1\right) \cdot -0.015625 + \left(\beta \cdot \beta\right) \cdot -0.078125}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 5.802994047758486 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2
Error	14.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 5.802994047758486 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Error	11.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 5.802994047758486 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Error	11.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i}}\\ \mathbf{elif}\;\beta \leq 5.802994047758486 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.200716879273244 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.4270341292843418 \cdot 10^{+110}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 6.404682674035876 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	16.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.404682674035876 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	57.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022306 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))