Octave 3.8, jcobi/4

Average Error: 54.2 → 10.1

Time: 26.7s

Precision: binary64

Cost: 27340

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i + \left(\beta + \alpha\right)}{t_0}\\ t_2 := \left(\beta + \alpha\right) + i \cdot 2\\ t_3 := \frac{i}{t_0}\\ \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(0.5 + \frac{\beta + \alpha}{i} \cdot -0.25\right) \cdot t_1\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{t_2 \cdot t_2 + -1}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \log \left(e^{t_3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\frac{\beta}{\alpha + i}} \cdot \frac{\beta + \left(\alpha + i\right)}{t_0}\\ \end{array} \]

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 (fma i 2.0 (+ beta alpha)))
        (t_1 (/ (+ i (+ beta alpha)) t_0))
        (t_2 (+ (+ beta alpha) (* i 2.0)))
        (t_3 (/ i t_0)))
   (if (<= beta 2.9e+73)
     (* (* (+ 0.5 (* (/ (+ beta alpha) i) -0.25)) t_1) 0.25)
     (if (<= beta 8e+86)
       (/
        (/ (* i i) (/ (pow (+ beta (* i 2.0)) 2.0) (pow (+ beta i) 2.0)))
        (+ (* t_2 t_2) -1.0))
       (if (<= beta 9.5e+186)
         (* 0.25 (* t_1 (log (exp t_3))))
         (* (/ t_3 (/ beta (+ alpha i))) (/ (+ 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 = fma(i, 2.0, (beta + alpha));
	double t_1 = (i + (beta + alpha)) / t_0;
	double t_2 = (beta + alpha) + (i * 2.0);
	double t_3 = i / t_0;
	double tmp;
	if (beta <= 2.9e+73) {
		tmp = ((0.5 + (((beta + alpha) / i) * -0.25)) * t_1) * 0.25;
	} else if (beta <= 8e+86) {
		tmp = ((i * i) / (pow((beta + (i * 2.0)), 2.0) / pow((beta + i), 2.0))) / ((t_2 * t_2) + -1.0);
	} else if (beta <= 9.5e+186) {
		tmp = 0.25 * (t_1 * log(exp(t_3)));
	} else {
		tmp = (t_3 / (beta / (alpha + i))) * ((beta + (alpha + i)) / 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 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(Float64(i + Float64(beta + alpha)) / t_0)
	t_2 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_3 = Float64(i / t_0)
	tmp = 0.0
	if (beta <= 2.9e+73)
		tmp = Float64(Float64(Float64(0.5 + Float64(Float64(Float64(beta + alpha) / i) * -0.25)) * t_1) * 0.25);
	elseif (beta <= 8e+86)
		tmp = Float64(Float64(Float64(i * i) / Float64((Float64(beta + Float64(i * 2.0)) ^ 2.0) / (Float64(beta + i) ^ 2.0))) / Float64(Float64(t_2 * t_2) + -1.0));
	elseif (beta <= 9.5e+186)
		tmp = Float64(0.25 * Float64(t_1 * log(exp(t_3))));
	else
		tmp = Float64(Float64(t_3 / Float64(beta / Float64(alpha + i))) * Float64(Float64(beta + Float64(alpha + i)) / 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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2.9e+73], N[(N[(N[(0.5 + N[(N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[beta, 8e+86], N[(N[(N[(i * i), $MachinePrecision] / N[(N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9.5e+186], N[(0.25 * N[(t$95$1 * N[Log[N[Exp[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + N[(alpha + i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.9000000000000002e73

   1. Initial program 48.0

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

   2. Simplified 31.3

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

      [Start] 48.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 48.1	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 31.3	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 2.1

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

   4. Taylor expanded in i around inf 2.2

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

   5. Simplified 2.2

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

      [Start] 2.2	\[ \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot 0.25 \]
      *-commutative [=>] 2.2	\[ \left(\left(0.5 + \color{blue}{\frac{\beta + \alpha}{i} \cdot -0.25}\right) \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot 0.25 \]

   if 2.9000000000000002e73 < beta < 8.0000000000000001e86

   1. Initial program 49.7

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

   2. Taylor expanded in alpha around 0 50.9

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

   3. Simplified 32.3

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

      [Start] 50.9	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 32.3	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 32.3	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 32.3	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if 8.0000000000000001e86 < beta < 9.49999999999999999e186

   1. Initial program 59.4

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

   2. Simplified 41.0

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

      [Start] 59.4	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 63.8	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 41.1	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 28.5

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

   4. Applied egg-rr 28.2

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

   if 9.49999999999999999e186 < 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. Simplified 56.5

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      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 [=>] 56.5	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 12.9

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

   4. Applied egg-rr 31.6

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

   5. Applied egg-rr 12.8

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(0.5 + \frac{\beta + \alpha}{i} \cdot -0.25\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \left(\frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \log \left(e^{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\frac{\beta}{\alpha + i}} \cdot \frac{\beta + \left(\alpha + i\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	15176

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := \frac{\beta + \alpha}{i}\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(0.5 + t_2 \cdot -0.25\right) \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + t_2 \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t_0}}{\frac{\beta}{\alpha + i}} \cdot \frac{\beta + \left(\alpha + i\right)}{t_0}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i + \left(\beta + \alpha\right)}{t_0} \cdot \frac{i}{t_0}\right) \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 3
Error	9.7
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t_0}}{\frac{\beta}{\alpha + i}} \cdot \frac{\beta + \left(\alpha + i\right)}{t_0}\\ \end{array} \]

Alternative 4
Error	9.7
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 5
Error	9.8
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	9.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	9.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\alpha + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	9.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	17.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+267}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 10
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 11
Error	17.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+269}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	57.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023054 
(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)))