Octave 3.8, jcobi/4

Average Accuracy: 15.7% → 84.0%

Time: 34.3s

Precision: binary64

Cost: 14852

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;i \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{{\left(\frac{i}{\frac{t_1}{i + \beta}}\right)}^{2}}{t_0 \cdot t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \end{array} \]

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 (+ (+ beta alpha) (* i 2.0))) (t_1 (fma i 2.0 (+ beta alpha))))
   (if (<= i 5e+153)
     (/ (pow (/ i (/ t_1 (+ i beta))) 2.0) (+ (* t_0 t_0) -1.0))
     (* (* (/ i t_1) (/ (+ i beta) (+ beta (* i 2.0)))) 0.25))))

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 = (beta + alpha) + (i * 2.0);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (i <= 5e+153) {
		tmp = pow((i / (t_1 / (i + beta))), 2.0) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = ((i / t_1) * ((i + beta) / (beta + (i * 2.0)))) * 0.25;
	}
	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(beta + alpha) + Float64(i * 2.0))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (i <= 5e+153)
		tmp = Float64((Float64(i / Float64(t_1 / Float64(i + beta))) ^ 2.0) / Float64(Float64(t_0 * t_0) + -1.0));
	else
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + beta) / Float64(beta + Float64(i * 2.0)))) * 0.25);
	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[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 5e+153], N[(N[Power[N[(i / N[(t$95$1 / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 5.00000000000000018e153

   1. Initial program 31.9%

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

   2. Taylor expanded in alpha around 0 31.6%

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

   3. Simplified 31.6%

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

      [Start] 31.6	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 31.6	\[ \frac{\frac{\color{blue}{\left(i \cdot i\right)} \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)}^{2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof

      [Start] 31.6	\[ \frac{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      expm1-log1p-u [=>] 29.4	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      expm1-udef [=>] 29.4	\[ \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 83.3%

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

      [Start] 69.9	\[ \frac{e^{\mathsf{log1p}\left({\left(\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)}^{2}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      expm1-def [=>] 69.9	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)}^{2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      expm1-log1p [=>] 75.8	\[ \frac{\color{blue}{{\left(\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 83.3	\[ \frac{{\color{blue}{\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \beta}}\right)}}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 83.3	\[ \frac{{\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\beta + i}}}\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if 5.00000000000000018e153 < i

   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. Simplified 0.1%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 0.1	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 84.8%

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

   4. Taylor expanded in alpha around 0 84.8%

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{\frac{\beta + i}{\beta + 2 \cdot i}}\right) \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{{\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \beta}}\right)}^{2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.1%
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+131}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{1 + t_0} \cdot \frac{i}{t_0 + -1}\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	8004

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\\ \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+131}:\\ \;\;\;\;t_0 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	7748

\[\begin{array}{l} t_0 := \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+131}:\\ \;\;\;\;\left(t_0 \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	85.2%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+131}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+131}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	85.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+131}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 7
Accuracy	82.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	82.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+198}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 9
Accuracy	74.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+236}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	10.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023126 
(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)))