Octave 3.8, jcobi/4

Average Error: 54.2 → 13.2

Time: 1.0min

Precision: binary64

Cost: 7888

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_2 := t_1 - \left(-0.0625 + t_1\right)\\ t_3 := \frac{\beta + i}{\alpha}\\ t_4 := i \cdot \left(\beta + i\right)\\ t_5 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\beta + i}{\frac{1}{i}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}}{t_5 \cdot t_5 - 1}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+226}:\\ \;\;\;\;\frac{t_3 \cdot i}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) + -1}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+243}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{elif}\;i \ne 0:\\ \;\;\;\;\frac{t_3}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{\alpha \cdot \alpha}\\ \end{array} \]

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 (/ (* 0.125 alpha) i))
        (t_1 (/ (* -0.0625 (+ beta alpha)) i))
        (t_2 (- t_1 (+ -0.0625 t_1)))
        (t_3 (/ (+ beta i) alpha))
        (t_4 (* i (+ beta i)))
        (t_5 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= alpha 9e+83)
     t_2
     (if (<= alpha 3.2e+114)
       (/ (if (!= i 0.0) (/ (+ beta i) (/ 1.0 i)) t_4) (- (* t_5 t_5) 1.0))
       (if (<= alpha 4.9e+148)
         t_2
         (if (<= alpha 1.3e+226)
           (/ (* t_3 i) (+ (+ alpha (fma 2.0 i beta)) -1.0))
           (if (<= alpha 5.2e+243)
             (- (+ 0.0625 t_0) t_0)
             (if (!= i 0.0) (/ t_3 (/ alpha i)) (/ t_4 (* alpha alpha))))))))))

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 = (0.125 * alpha) / i;
	double t_1 = (-0.0625 * (beta + alpha)) / i;
	double t_2 = t_1 - (-0.0625 + t_1);
	double t_3 = (beta + i) / alpha;
	double t_4 = i * (beta + i);
	double t_5 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (alpha <= 9e+83) {
		tmp = t_2;
	} else if (alpha <= 3.2e+114) {
		double tmp_1;
		if (i != 0.0) {
			tmp_1 = (beta + i) / (1.0 / i);
		} else {
			tmp_1 = t_4;
		}
		tmp = tmp_1 / ((t_5 * t_5) - 1.0);
	} else if (alpha <= 4.9e+148) {
		tmp = t_2;
	} else if (alpha <= 1.3e+226) {
		tmp = (t_3 * i) / ((alpha + fma(2.0, i, beta)) + -1.0);
	} else if (alpha <= 5.2e+243) {
		tmp = (0.0625 + t_0) - t_0;
	} else if (i != 0.0) {
		tmp = t_3 / (alpha / i);
	} else {
		tmp = t_4 / (alpha * alpha);
	}
	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(0.125 * alpha) / i)
	t_1 = Float64(Float64(-0.0625 * Float64(beta + alpha)) / i)
	t_2 = Float64(t_1 - Float64(-0.0625 + t_1))
	t_3 = Float64(Float64(beta + i) / alpha)
	t_4 = Float64(i * Float64(beta + i))
	t_5 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 9e+83)
		tmp = t_2;
	elseif (alpha <= 3.2e+114)
		tmp_1 = 0.0
		if (i != 0.0)
			tmp_1 = Float64(Float64(beta + i) / Float64(1.0 / i));
		else
			tmp_1 = t_4;
		end
		tmp = Float64(tmp_1 / Float64(Float64(t_5 * t_5) - 1.0));
	elseif (alpha <= 4.9e+148)
		tmp = t_2;
	elseif (alpha <= 1.3e+226)
		tmp = Float64(Float64(t_3 * i) / Float64(Float64(alpha + fma(2.0, i, beta)) + -1.0));
	elseif (alpha <= 5.2e+243)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	elseif (i != 0.0)
		tmp = Float64(t_3 / Float64(alpha / i));
	else
		tmp = Float64(t_4 / Float64(alpha * alpha));
	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[(0.125 * alpha), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.0625 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(-0.0625 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + i), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 9e+83], t$95$2, If[LessEqual[alpha, 3.2e+114], N[(If[Unequal[i, 0.0], N[(N[(beta + i), $MachinePrecision] / N[(1.0 / i), $MachinePrecision]), $MachinePrecision], t$95$4] / N[(N[(t$95$5 * t$95$5), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 4.9e+148], t$95$2, If[LessEqual[alpha, 1.3e+226], N[(N[(t$95$3 * i), $MachinePrecision] / N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, 5.2e+243], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], If[Unequal[i, 0.0], N[(t$95$3 / N[(alpha / i), $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if alpha < 8.9999999999999999e83 or 3.2e114 < alpha < 4.9e148

   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. Taylor expanded in i around inf 42.9

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

   3. Taylor expanded in i around inf 9.9

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

   4. Simplified 9.9

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

   5. Applied egg-rr 9.9

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

   if 8.9999999999999999e83 < alpha < 3.2e114

   1. Initial program 54.8

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

   2. Taylor expanded in alpha around inf 46.2

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

   3. Applied egg-rr 46.2

      \[\leadsto \frac{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\beta + i}{\frac{1}{i}}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\beta + i\right)\\ } \end{array}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if 4.9e148 < alpha < 1.3000000000000001e226

   1. Initial program 63.9

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

   2. Taylor expanded in alpha around inf 56.0

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

   3. Applied egg-rr 24.8

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

   4. Taylor expanded in alpha around inf 27.7

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

   if 1.3000000000000001e226 < alpha < 5.19999999999999993e243

   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. Simplified 64.0

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

   3. Taylor expanded in i around inf 36.0

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

   4. Simplified 36.0

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

   5. Taylor expanded in beta around 0 36.6

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

   6. Simplified 36.6

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

   7. Taylor expanded in beta around 0 36.0

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

   8. Simplified 36.0

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

   if 5.19999999999999993e243 < alpha

   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. Simplified 64.0

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

   3. Taylor expanded in alpha around inf 42.6

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

   4. Applied egg-rr 37.2

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

   5. Applied egg-rr 11.3

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ } \end{array}} \]
3. Recombined 5 regimes into one program.

Alternatives

Alternative 1
Error	10.4
Cost	38340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(\beta \cdot \alpha + t_3\right)}{t_1}}{t_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{t_3}{t_4} \cdot \frac{\frac{\mathsf{fma}\left(\beta, \alpha, t_3\right)}{t_4}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}}{1 + t_4}\\ \mathbf{else}:\\ \;\;\;\;t_2 - \left(-0.0625 + t_2\right)\\ \end{array} \]

Alternative 2
Error	10.5
Cost	38020

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(\beta \cdot \alpha + t_3\right)}{t_1}}{t_1 - 1} \leq \infty:\\ \;\;\;\;\left(\frac{-1}{1 - {t_4}^{2}} \cdot \frac{t_3}{t_4}\right) \cdot \frac{\mathsf{fma}\left(\beta, \alpha, t_3\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;t_2 - \left(-0.0625 + t_2\right)\\ \end{array} \]

Alternative 3
Error	13.5
Cost	14540

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_2 := \alpha + \mathsf{fma}\left(2, i, \beta\right)\\ t_3 := t_2 + -1\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_1 - \left(-0.0625 + t_1\right)\\ \mathbf{elif}\;\alpha \leq 4.3 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{i}{\left(1 + \alpha\right) + \left(i + i\right)} \cdot i}{t_3}\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + i}{1 + t_2} \cdot i}{t_3}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7880

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_1 - \left(-0.0625 + t_1\right)\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{i}{\left(1 + \alpha\right) + \left(i + i\right)} \cdot i}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) + -1}\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+242}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{elif}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 5
Error	13.2
Cost	1996

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_2 := i \cdot \left(\beta + i\right)\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_4 := t_1 - \left(-0.0625 + t_1\right)\\ t_5 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\alpha \cdot \alpha}\\ \end{array}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\alpha \leq 4.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{\begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\beta + i}{\frac{1}{i}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}}{t_3 \cdot t_3 - 1}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+150}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\alpha \leq 1.25 \cdot 10^{+226}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+242}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 6
Error	13.1
Cost	1608

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_2 := t_1 - \left(-0.0625 + t_1\right)\\ t_3 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array}\\ t_4 := \frac{0.125 \cdot \alpha}{i}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{+114}:\\ \;\;\;\;\frac{i \cdot i}{t_0 \cdot t_0 - 1}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\alpha \leq 8.4 \cdot 10^{+242}:\\ \;\;\;\;\left(0.0625 + t_4\right) - t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	14.8
Cost	1492

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \left(0.0625 + t_0\right) - t_0\\ t_2 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\alpha \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;i \cdot \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(\beta + i\right)\right)\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 2.1 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+242}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	12.9
Cost	1492

\[\begin{array}{l} t_0 := \frac{0.125 \cdot \alpha}{i}\\ t_1 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array}\\ t_2 := \left(0.0625 + t_0\right) - t_0\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;\frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i} - \left(-0.0625 + \frac{-0.0625 \cdot \beta}{i}\right)\\ \mathbf{elif}\;\alpha \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;i \cdot \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(\beta + i\right)\right)\\ \mathbf{elif}\;\alpha \leq 1.18 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{+242}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	12.9
Cost	1492

\[\begin{array}{l} t_0 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_1 := t_0 - \left(-0.0625 + t_0\right)\\ t_2 := \frac{0.125 \cdot \alpha}{i}\\ t_3 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;i \cdot \left(\frac{1}{\alpha \cdot \alpha} \cdot \left(\beta + i\right)\right)\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.25 \cdot 10^{+226}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\alpha \leq 8.4 \cdot 10^{+242}:\\ \;\;\;\;\left(0.0625 + t_2\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	12.9
Cost	1492

\[\begin{array}{l} t_0 := \frac{-0.0625 \cdot \left(\beta + \alpha\right)}{i}\\ t_1 := t_0 - \left(-0.0625 + t_0\right)\\ t_2 := \begin{array}{l} \mathbf{if}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array}\\ t_3 := \frac{0.125 \cdot \alpha}{i}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(1 + \alpha\right) + \left(i + i\right)\right) \cdot \left(\left(\alpha + \left(i + i\right)\right) - 1\right)}\\ \mathbf{elif}\;\alpha \leq 5.4 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+242}:\\ \;\;\;\;\left(0.0625 + t_3\right) - t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	18.7
Cost	844

\[\begin{array}{l} t_0 := i \cdot \frac{\frac{i}{\alpha}}{\alpha}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	19.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\alpha \leq 9 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\alpha}}{\alpha}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\alpha}}{\alpha}\\ \end{array} \]

Alternative 13
Error	15.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;i \ne 0:\\ \;\;\;\;\frac{\frac{\beta + i}{\alpha}}{\frac{\alpha}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\beta + i\right)}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 14
Error	15.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+83}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + i}{\alpha} \cdot \frac{i}{\alpha}\\ \end{array} \]

Alternative 15
Error	18.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.4 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{\beta}{\alpha}}{\alpha}\\ \end{array} \]

Alternative 16
Error	18.7
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023010 
(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)))