Octave 3.8, jcobi/4

Average Error: 54.2 → 10.5

Time: 26.3s

Precision: binary64

Cost: 8012

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\beta + \left(\alpha + 1\right)}\\ \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.8584219890022636 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.0645449867813709 \cdot 10^{+250}:\\ \;\;\;\;\left(\frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)} \cdot \left(0.5 + -0.25 \cdot \frac{\beta}{i}\right)\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot -2 + 2 \cdot \left(1 - \left(\beta + \alpha\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0
         (*
          (/ (+ i alpha) (+ (fma i 2.0 (+ beta alpha)) -1.0))
          (/ i (+ beta (+ alpha 1.0))))))
   (if (<= beta 2.320133153966531e+178)
     (+ 0.0625 (/ 0.015625 (* i i)))
     (if (<= beta 2.8584219890022636e+218)
       t_0
       (if (<= beta 1.0645449867813709e+250)
         (*
          (*
           (/ (+ beta i) (+ (* i 2.0) (+ beta 1.0)))
           (+ 0.5 (* -0.25 (/ beta i))))
          (+
           (+ 0.25 (* 0.25 (/ (+ beta alpha) i)))
           (*
            0.0625
            (/ (+ (* (+ beta alpha) -2.0) (* 2.0 (- 1.0 (+ beta alpha)))) i))))
         t_0)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = ((i + alpha) / (fma(i, 2.0, (beta + alpha)) + -1.0)) * (i / (beta + (alpha + 1.0)));
	double tmp;
	if (beta <= 2.320133153966531e+178) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else if (beta <= 2.8584219890022636e+218) {
		tmp = t_0;
	} else if (beta <= 1.0645449867813709e+250) {
		tmp = (((beta + i) / ((i * 2.0) + (beta + 1.0))) * (0.5 + (-0.25 * (beta / i)))) * ((0.25 + (0.25 * ((beta + alpha) / i))) + (0.0625 * ((((beta + alpha) * -2.0) + (2.0 * (1.0 - (beta + alpha)))) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(i + alpha) / Float64(fma(i, 2.0, Float64(beta + alpha)) + -1.0)) * Float64(i / Float64(beta + Float64(alpha + 1.0))))
	tmp = 0.0
	if (beta <= 2.320133153966531e+178)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	elseif (beta <= 2.8584219890022636e+218)
		tmp = t_0;
	elseif (beta <= 1.0645449867813709e+250)
		tmp = Float64(Float64(Float64(Float64(beta + i) / Float64(Float64(i * 2.0) + Float64(beta + 1.0))) * Float64(0.5 + Float64(-0.25 * Float64(beta / i)))) * Float64(Float64(0.25 + Float64(0.25 * Float64(Float64(beta + alpha) / i))) + Float64(0.0625 * Float64(Float64(Float64(Float64(beta + alpha) * -2.0) + Float64(2.0 * Float64(1.0 - Float64(beta + alpha)))) / i))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(beta + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.320133153966531e+178], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.8584219890022636e+218], t$95$0, If[LessEqual[beta, 1.0645449867813709e+250], N[(N[(N[(N[(beta + i), $MachinePrecision] / N[(N[(i * 2.0), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 + N[(0.25 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[(N[(N[(N[(beta + alpha), $MachinePrecision] * -2.0), $MachinePrecision] + N[(2.0 * N[(1.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.320133153966531e178

   1. Initial program 50.7

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

   2. Taylor expanded in i around inf 40.0

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Simplified 40.0

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (*.f64 (*.f64 i i) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) 1/4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (pow.f64 i 2))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in i around inf 40.5

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{4 \cdot {i}^{2}} - 1} \]

   5. Simplified 40.5

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]
      Proof
      (*.f64 (*.f64 i i) 4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) 4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 4 (pow.f64 i 2))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in i around inf 7.6

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]

   7. Simplified 7.6

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
      Proof
      (+.f64 1/16 (/.f64 1/64 (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (/.f64 (Rewrite<= metadata-eval (*.f64 1/64 1)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (/.f64 (*.f64 1/64 1) (Rewrite<= unpow2_binary64 (pow.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (Rewrite<= associate-*r/_binary64 (*.f64 1/64 (/.f64 1 (pow.f64 i 2))))): 0 points increase in error, 0 points decrease in error

   if 2.320133153966531e178 < beta < 2.85842198900226364e218 or 1.06454498678137085e250 < beta

   1. Initial program 64.0

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

   2. Applied egg-rr 54.5

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

   3. Taylor expanded in beta around inf 13.2

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

   4. Taylor expanded in i around 0 14.7

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

   5. Simplified 14.7

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

   if 2.85842198900226364e218 < beta < 1.06454498678137085e250

   1. Initial program 64.0

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

   2. Applied egg-rr 51.5

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

   3. Taylor expanded in i around inf 32.1

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) + 2 \cdot \left(\left(\beta + \alpha\right) - 1\right)}{i}\right)} \]

   4. Taylor expanded in alpha around 0 51.0

      \[\leadsto \color{blue}{\frac{i \cdot \left(\beta + i\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)}} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) + 2 \cdot \left(\left(\beta + \alpha\right) - 1\right)}{i}\right) \]

   5. Simplified 32.1

      \[\leadsto \color{blue}{\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\left(\beta + 1\right) + i \cdot 2}\right)} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) + 2 \cdot \left(\left(\beta + \alpha\right) - 1\right)}{i}\right) \]
      Proof
      (*.f64 (/.f64 i (+.f64 beta (*.f64 i 2))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 beta (*.f64 2 i))) (/.f64 (+.f64 beta i) (+.f64 (+.f64 beta 1) (Rewrite<= *-commutative_binary64 (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 beta (*.f64 2 i))) (/.f64 (+.f64 beta i) (Rewrite<= associate-+r+_binary64 (+.f64 beta (+.f64 1 (*.f64 2 i)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 beta i)) (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 1 (*.f64 2 i)))))): 158 points increase in error, 14 points decrease in error

   6. Taylor expanded in i around inf 32.2

      \[\leadsto \left(\color{blue}{\left(0.5 + -0.25 \cdot \frac{\beta}{i}\right)} \cdot \frac{\beta + i}{\left(\beta + 1\right) + i \cdot 2}\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right) + 2 \cdot \left(\left(\beta + \alpha\right) - 1\right)}{i}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.8584219890022636 \cdot 10^{+218}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\beta + \left(\alpha + 1\right)}\\ \mathbf{elif}\;\beta \leq 1.0645449867813709 \cdot 10^{+250}:\\ \;\;\;\;\left(\frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)} \cdot \left(0.5 + -0.25 \cdot \frac{\beta}{i}\right)\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot -2 + 2 \cdot \left(1 - \left(\beta + \alpha\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\beta + \left(\alpha + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	22212

\[\begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_1}{i + \left(\beta + \alpha\right)}}}{t_1 + 1} \cdot \frac{\left(0.5 \cdot \left(\beta + \alpha\right) + i \cdot 0.5\right) + \left(\beta + \alpha\right) \cdot -0.25}{t_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \frac{1 + t_0}{\beta + i}}}{\frac{-1 + t_0}{i + \alpha}}\\ \end{array} \]

Alternative 2
Error	9.1
Cost	21188

\[\begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) \cdot \frac{1 + t_0}{\beta + i}}}{\frac{-1 + t_0}{i + \alpha}}\\ \end{array} \]

Alternative 3
Error	9.2
Cost	8516

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]

Alternative 4
Error	10.5
Cost	7756

\[\begin{array}{l} t_0 := \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\beta}\\ \mathbf{if}\;\beta \leq 2.320133153966531 \cdot 10^{+178}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.8584219890022636 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.0645449867813709 \cdot 10^{+250}:\\ \;\;\;\;\left(\frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)} \cdot \left(0.5 + -0.25 \cdot \frac{\beta}{i}\right)\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot -2 + 2 \cdot \left(1 - \left(\beta + \alpha\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.6
Cost	3404

\[\begin{array}{l} t_0 := \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.8584219890022636 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.0645449867813709 \cdot 10^{+250}:\\ \;\;\;\;\left(\frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)} \cdot \left(0.5 + -0.25 \cdot \frac{\beta}{i}\right)\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot -2 + 2 \cdot \left(1 - \left(\beta + \alpha\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	10.8
Cost	2764

\[\begin{array}{l} t_0 := \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.8584219890022636 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.0645449867813709 \cdot 10^{+250}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \left(\left(0.25 + 0.25 \cdot \frac{\beta + \alpha}{i}\right) + 0.0625 \cdot \left(\beta \cdot \frac{-4}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Error	9.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	10.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	10.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6700007595078876 \cdot 10^{+184}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 11
Error	18.4
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022308 
(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)))