Octave 3.8, jcobi/4

Average Accuracy: 16.0% → 97.6%

Time: 33.8s

Precision: binary64

Cost: 21572

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := 1 + t_0\\ \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+169}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot \left(\frac{i}{\beta + \mathsf{fma}\left(i, 2, -1\right)} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\alpha + \beta\right)}}}{t_1} \cdot \frac{\alpha + i}{-1 + t_0}\\ \end{array} \]

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 (+ alpha beta))) (t_1 (+ 1.0 t_0)))
   (if (<= alpha 4.3e+169)
     (/
      (*
       (* i (/ (+ i beta) (+ beta (* i 2.0))))
       (* (/ i (+ beta (fma i 2.0 -1.0))) (/ (+ i beta) (fma i 2.0 beta))))
      t_1)
     (*
      (/ (/ i (/ t_0 (+ i (+ alpha beta)))) t_1)
      (/ (+ alpha i) (+ -1.0 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, (alpha + beta));
	double t_1 = 1.0 + t_0;
	double tmp;
	if (alpha <= 4.3e+169) {
		tmp = ((i * ((i + beta) / (beta + (i * 2.0)))) * ((i / (beta + fma(i, 2.0, -1.0))) * ((i + beta) / fma(i, 2.0, beta)))) / t_1;
	} else {
		tmp = ((i / (t_0 / (i + (alpha + beta)))) / t_1) * ((alpha + i) / (-1.0 + 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(alpha + beta))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (alpha <= 4.3e+169)
		tmp = Float64(Float64(Float64(i * Float64(Float64(i + beta) / Float64(beta + Float64(i * 2.0)))) * Float64(Float64(i / Float64(beta + fma(i, 2.0, -1.0))) * Float64(Float64(i + beta) / fma(i, 2.0, beta)))) / t_1);
	else
		tmp = Float64(Float64(Float64(i / Float64(t_0 / Float64(i + Float64(alpha + beta)))) / t_1) * Float64(Float64(alpha + i) / Float64(-1.0 + 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[alpha, 4.3e+169], N[(N[(N[(i * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(beta + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(i / N[(t$95$0 / N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.3000000000000001e169

   1. Initial program 16.4%

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

   2. Applied egg-rr 42.7%

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

      [Start] 16.4	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 39.3	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 39.3	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 42.7	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in alpha around 0 39.1%

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

   4. Applied egg-rr 98.2%

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

      [Start] 39.1	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(\beta + i\right)}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
      associate-*l/ [=>] 39.1	\[ \color{blue}{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}} \cdot \frac{i \cdot \left(\beta + i\right)}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}} \]

   5. Taylor expanded in alpha around 0 98.2%

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

   if 4.3000000000000001e169 < alpha

   1. Initial program 0.0%

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

   2. Applied egg-rr 0.0%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 0.0	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 0.0	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 75.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+169}:\\ \;\;\;\;\frac{\left(i \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot \left(\frac{i}{\beta + \mathsf{fma}\left(i, 2, -1\right)} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + i}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	21444

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := 1 + t_0\\ t_2 := -1 + t_0\\ t_3 := \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+169}:\\ \;\;\;\;\frac{t_3}{t_1} \cdot \frac{t_3}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\alpha + \beta\right)}}}{t_1} \cdot \frac{\alpha + i}{t_2}\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	15428

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := 1 + t_0\\ t_2 := \frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+169}:\\ \;\;\;\;\frac{t_2}{t_1} \cdot \frac{t_2}{-1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\alpha + \beta\right)}}}{t_1} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	14660

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_1}{i + \beta}}}{-1 + t_0} \cdot \left(\frac{i}{t_1} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\alpha + \beta\right)}}}{1 + t_0} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	97.6%
Cost	9028

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \beta}}}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{i \cdot 2 + \left(\beta + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	85.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+156}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	83.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+156}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	83.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+156}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Accuracy	74.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	9.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023146 
(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)))