Octave 3.8, jcobi/4

Average Error: 53.8 → 10.8

Time: 29.1s

Precision: binary64

Cost: 41288

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

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

Math

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

↓

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \frac{i}{t_1}\\ t_3 := \left(t_2 \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ t_4 := t_2 \cdot \frac{t_0}{t_1}\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;t_4 \cdot \frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t_1, t_1, -1\right)}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+192} \lor \neg \left(\beta \leq 8 \cdot 10^{+217}\right):\\ \;\;\;\;t_4 \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

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
 (let* ((t_0 (+ i (+ beta alpha)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (/ i t_1))
        (t_3 (* (* t_2 (/ (+ beta i) (+ beta (* i 2.0)))) 0.25))
        (t_4 (* t_2 (/ t_0 t_1))))
   (if (<= beta 6.5e+81)
     t_3
     (if (<= beta 3.8e+110)
       (* t_4 (/ (fma i t_0 (* beta alpha)) (fma t_1 t_1 -1.0)))
       (if (<= beta 3.8e+135)
         t_3
         (if (or (<= beta 1.65e+192) (not (<= beta 8e+217)))
           (* t_4 (/ (+ i alpha) beta))
           (+ (+ 0.0625 (* 0.125 (/ beta i))) (/ -0.125 (/ 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 t_0 = i + (beta + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = i / t_1;
	double t_3 = (t_2 * ((beta + i) / (beta + (i * 2.0)))) * 0.25;
	double t_4 = t_2 * (t_0 / t_1);
	double tmp;
	if (beta <= 6.5e+81) {
		tmp = t_3;
	} else if (beta <= 3.8e+110) {
		tmp = t_4 * (fma(i, t_0, (beta * alpha)) / fma(t_1, t_1, -1.0));
	} else if (beta <= 3.8e+135) {
		tmp = t_3;
	} else if ((beta <= 1.65e+192) || !(beta <= 8e+217)) {
		tmp = t_4 * ((i + alpha) / beta);
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + (-0.125 / (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)
	t_0 = Float64(i + Float64(beta + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(i / t_1)
	t_3 = Float64(Float64(t_2 * Float64(Float64(beta + i) / Float64(beta + Float64(i * 2.0)))) * 0.25)
	t_4 = Float64(t_2 * Float64(t_0 / t_1))
	tmp = 0.0
	if (beta <= 6.5e+81)
		tmp = t_3;
	elseif (beta <= 3.8e+110)
		tmp = Float64(t_4 * Float64(fma(i, t_0, Float64(beta * alpha)) / fma(t_1, t_1, -1.0)));
	elseif (beta <= 3.8e+135)
		tmp = t_3;
	elseif ((beta <= 1.65e+192) || !(beta <= 8e+217))
		tmp = Float64(t_4 * Float64(Float64(i + alpha) / beta));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(-0.125 / 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_] := Block[{t$95$0 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(N[(beta + i), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.5e+81], t$95$3, If[LessEqual[beta, 3.8e+110], N[(t$95$4 * N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.8e+135], t$95$3, If[Or[LessEqual[beta, 1.65e+192], N[Not[LessEqual[beta, 8e+217]], $MachinePrecision]], N[(t$95$4 * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 / N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 6.4999999999999996e81 or 3.79999999999999989e110 < beta < 3.8000000000000001e135

   1. Initial program 48.3

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

   2. Simplified 31.7

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

      [Start] 48.3	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 49.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 31.7	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 4.1

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

   4. Taylor expanded in alpha around 0 4.2

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

   if 6.4999999999999996e81 < beta < 3.79999999999999989e110

   1. Initial program 53.2

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

   2. Simplified 31.7

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

      [Start] 53.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 63.5	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 31.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   if 3.8000000000000001e135 < beta < 1.65000000000000005e192 or 7.99999999999999968e217 < 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. Simplified 55.4

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

      [Start] 63.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 55.4	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 17.5

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

   if 1.65000000000000005e192 < beta < 7.99999999999999968e217

   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 54.6

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 54.6	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 31.1

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

   4. Taylor expanded in beta around inf 31.1

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

   5. Simplified 31.1

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

      [Start] 31.1	\[ \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      associate-*r/ [=>] 31.1	\[ \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   6. Taylor expanded in i around 0 31.1

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

   7. Taylor expanded in beta around inf 31.1

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

   8. Simplified 32.8

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

      [Start] 31.1	\[ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]
      associate-*r/ [=>] 31.1	\[ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{0.125 \cdot \beta}{i}} \]
      associate-/l* [=>] 32.8	\[ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{0.125}{\frac{i}{\beta}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+192} \lor \neg \left(\beta \leq 8 \cdot 10^{+217}\right):\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	15061

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := i \cdot \left(\alpha + \left(\beta + i\right)\right)\\ t_3 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_4 := \frac{i}{t_3}\\ t_5 := \left(t_4 \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ t_6 := \alpha + t_0\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+84}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{t_2}{t_6 \cdot t_6} \cdot \left(\beta \cdot \alpha + t_2\right)}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+136}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+217}\right):\\ \;\;\;\;\left(t_4 \cdot \frac{i + \left(\beta + \alpha\right)}{t_3}\right) \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 2
Error	10.8
Cost	8012

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ t_3 := i \cdot \left(\alpha + \left(\beta + i\right)\right)\\ t_4 := \alpha + t_0\\ \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{t_3}{t_4 \cdot t_4} \cdot \left(\beta \cdot \alpha + t_3\right)}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+192} \lor \neg \left(\beta \leq 4.4 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 3
Error	10.7
Cost	3656

\[\begin{array}{l} t_0 := \alpha + \left(\beta + i \cdot 2\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := i \cdot \left(\alpha + \left(\beta + i\right)\right)\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{t_2}{t_0 \cdot t_0} \cdot \left(\beta \cdot \alpha + t_2\right)}{-1 + t_1 \cdot t_1}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.5 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 4
Error	11.2
Cost	1736

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 2.35 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+191} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 5
Error	11.4
Cost	1608

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+84}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{-1 + \left(\beta \cdot \beta + 2 \cdot \left(\beta \cdot \left(\alpha + i \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 6
Error	11.2
Cost	1493

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+112}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.1 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{-0.125}{\frac{i}{\beta}}\\ \end{array} \]

Alternative 7
Error	11.5
Cost	1237

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 8.6 \cdot 10^{+193} \lor \neg \left(\beta \leq 1.02 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 8
Error	12.8
Cost	1109

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.65 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+193} \lor \neg \left(\beta \leq 1.02 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 9
Error	12.9
Cost	1109

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+193} \lor \neg \left(\beta \leq 1.02 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 10
Error	15.1
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.76 \cdot 10^{+195} \lor \neg \left(\beta \leq 1.3 \cdot 10^{+216}\right):\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 11
Error	11.7
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+195} \lor \neg \left(\beta \leq 1.02 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 12
Error	16.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	16.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+241}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 14
Error	18.6
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023060 
(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)))