Octave 3.8, jcobi/4

Average Error: 54.4 → 10.2

Time: 26.1s

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

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

↓

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \frac{t_0}{t_1}\\ t_3 := {\left(\beta + i\right)}^{2}\\ t_4 := \frac{i}{t_1}\\ \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot t_2\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+86}:\\ \;\;\;\;\left(t_2 \cdot t_4\right) \cdot \frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t_1, t_1, -1\right)}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot t_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{\beta + i} + \left(\alpha \cdot \left(\frac{1}{\beta + i} + \left(\frac{i}{t_3} \cdot -2 - \frac{\beta}{t_3}\right)\right) + 2 \cdot \frac{i}{\beta + i}\right)\right) \cdot \frac{t_1}{i}} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

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 (/ t_0 t_1))
        (t_3 (pow (+ beta i) 2.0))
        (t_4 (/ i t_1)))
   (if (<= beta 2.55e+67)
     (* (* (+ 0.5 (* -0.25 (/ (+ beta alpha) i))) t_2) 0.25)
     (if (<= beta 3.9e+86)
       (* (* t_2 t_4) (/ (fma i t_0 (* beta alpha)) (fma t_1 t_1 -1.0)))
       (if (<= beta 5.2e+117)
         (* 0.25 (* 0.5 t_4))
         (*
          (/
           1.0
           (*
            (+
             (/ beta (+ beta i))
             (+
              (*
               alpha
               (+ (/ 1.0 (+ beta i)) (- (* (/ i t_3) -2.0) (/ beta t_3))))
              (* 2.0 (/ i (+ beta i)))))
            (/ t_1 i)))
          (/ (+ alpha 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 = t_0 / t_1;
	double t_3 = pow((beta + i), 2.0);
	double t_4 = i / t_1;
	double tmp;
	if (beta <= 2.55e+67) {
		tmp = ((0.5 + (-0.25 * ((beta + alpha) / i))) * t_2) * 0.25;
	} else if (beta <= 3.9e+86) {
		tmp = (t_2 * t_4) * (fma(i, t_0, (beta * alpha)) / fma(t_1, t_1, -1.0));
	} else if (beta <= 5.2e+117) {
		tmp = 0.25 * (0.5 * t_4);
	} else {
		tmp = (1.0 / (((beta / (beta + i)) + ((alpha * ((1.0 / (beta + i)) + (((i / t_3) * -2.0) - (beta / t_3)))) + (2.0 * (i / (beta + i))))) * (t_1 / i))) * ((alpha + 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(t_0 / t_1)
	t_3 = Float64(beta + i) ^ 2.0
	t_4 = Float64(i / t_1)
	tmp = 0.0
	if (beta <= 2.55e+67)
		tmp = Float64(Float64(Float64(0.5 + Float64(-0.25 * Float64(Float64(beta + alpha) / i))) * t_2) * 0.25);
	elseif (beta <= 3.9e+86)
		tmp = Float64(Float64(t_2 * t_4) * Float64(fma(i, t_0, Float64(beta * alpha)) / fma(t_1, t_1, -1.0)));
	elseif (beta <= 5.2e+117)
		tmp = Float64(0.25 * Float64(0.5 * t_4));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(beta / Float64(beta + i)) + Float64(Float64(alpha * Float64(Float64(1.0 / Float64(beta + i)) + Float64(Float64(Float64(i / t_3) * -2.0) - Float64(beta / t_3)))) + Float64(2.0 * Float64(i / Float64(beta + i))))) * Float64(t_1 / i))) * Float64(Float64(alpha + 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[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(i / t$95$1), $MachinePrecision]}, If[LessEqual[beta, 2.55e+67], N[(N[(N[(0.5 + N[(-0.25 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[beta, 3.9e+86], N[(N[(t$95$2 * t$95$4), $MachinePrecision] * 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, 5.2e+117], N[(0.25 * N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(beta / N[(beta + i), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha * N[(N[(1.0 / N[(beta + i), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i / t$95$3), $MachinePrecision] * -2.0), $MachinePrecision] - N[(beta / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.5500000000000001e67

   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 33.1

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      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* [<=] 48.4	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 33.1	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 1.8

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

   4. Taylor expanded in i around inf 1.9

      \[\leadsto \left(\color{blue}{\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot 0.25 \]

   if 2.5500000000000001e67 < beta < 3.9000000000000002e86

   1. Initial program 50.5

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

   2. Simplified 32.8

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

      [Start] 50.5	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 57.2	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 32.9	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   if 3.9000000000000002e86 < beta < 5.1999999999999999e117

   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 34.0

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      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 [=>] 34.1	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 19.6

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

   4. Taylor expanded in i around inf 20.2

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{0.5}\right) \cdot 0.25 \]

   if 5.1999999999999999e117 < beta

   1. Initial program 63.5

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

   2. Simplified 52.8

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

      [Start] 63.5	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      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 [=>] 52.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 19.0

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

   4. Applied egg-rr 19.0

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

   5. Taylor expanded in alpha around 0 19.0

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{\beta}{\beta + i} + \left(\left(\frac{1}{\beta + i} - \left(\frac{\beta}{{\left(\beta + i\right)}^{2}} + 2 \cdot \frac{i}{{\left(\beta + i\right)}^{2}}\right)\right) \cdot \alpha + 2 \cdot \frac{i}{\beta + i}\right)\right)} \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i}} \cdot \frac{i + \alpha}{\beta} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+86}:\\ \;\;\;\;\left(\frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right), \mathsf{fma}\left(i, 2, \beta + \alpha\right), -1\right)}\\ \mathbf{elif}\;\beta \leq 5.2 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{\beta + i} + \left(\alpha \cdot \left(\frac{1}{\beta + i} + \left(\frac{i}{{\left(\beta + i\right)}^{2}} \cdot -2 - \frac{\beta}{{\left(\beta + i\right)}^{2}}\right)\right) + 2 \cdot \frac{i}{\beta + i}\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i}} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	22724

\[\begin{array}{l} t_0 := {\left(\beta + i\right)}^{2}\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_1} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\beta}{\beta + i} + \left(\alpha \cdot \left(\frac{1}{\beta + i} + \left(\frac{i}{t_0} \cdot -2 - \frac{\beta}{t_0}\right)\right) + 2 \cdot \frac{i}{\beta + i}\right)\right) \cdot \frac{t_1}{i}} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 2
Error	9.6
Cost	8132

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_1} \cdot \frac{\beta + i}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{1}{\frac{t_1}{i} \cdot \frac{t_0}{\beta + i}}\\ \end{array} \]

Alternative 3
Error	9.6
Cost	7748

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{1}{\frac{t_0}{i}}\\ \end{array} \]

Alternative 4
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{1}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i}}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 6
Error	9.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + i\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 7
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	17.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+273}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	57.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023059 
(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)))