Octave 3.8, jcobi/4

Average Error: 54.5 → 11.3

Time: 28.4s

Precision: binary64

Cost: 14797

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106} \lor \neg \left(\beta \leq 6.6 \cdot 10^{+124}\right) \land \beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \left(\beta + \alpha\right)}{t_1}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

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 (+ beta (* i 2.0))) (t_1 (fma i 2.0 (+ beta alpha))))
   (if (or (<= beta 3.3e+106)
           (and (not (<= beta 6.6e+124)) (<= beta 8.8e+223)))
     (* (* (/ i t_0) (/ (+ beta i) t_0)) 0.25)
     (* (* (/ i t_1) (/ (+ i (+ beta alpha)) t_1)) (/ (+ i alpha) 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 = beta + (i * 2.0);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if ((beta <= 3.3e+106) || (!(beta <= 6.6e+124) && (beta <= 8.8e+223))) {
		tmp = ((i / t_0) * ((beta + i) / t_0)) * 0.25;
	} else {
		tmp = ((i / t_1) * ((i + (beta + alpha)) / t_1)) * ((i + alpha) / 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(beta + Float64(i * 2.0))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if ((beta <= 3.3e+106) || (!(beta <= 6.6e+124) && (beta <= 8.8e+223)))
		tmp = Float64(Float64(Float64(i / t_0) * Float64(Float64(beta + i) / t_0)) * 0.25);
	else
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + Float64(beta + alpha)) / t_1)) * Float64(Float64(i + alpha) / 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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[beta, 3.3e+106], And[N[Not[LessEqual[beta, 6.6e+124]], $MachinePrecision], LessEqual[beta, 8.8e+223]]], N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.30000000000000008e106 or 6.60000000000000029e124 < beta < 8.7999999999999999e223

   1. Initial program 52.3

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

   2. Simplified 36.7

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

      [Start] 52.3	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 53.2	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 36.7	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 10.3

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

   4. Taylor expanded in alpha around 0 10.3

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

   5. Taylor expanded in alpha around 0 10.3

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

   6. Simplified 10.3

      \[\leadsto \left(\frac{i}{\color{blue}{\beta + i \cdot 2}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]
      Proof

      [Start] 10.3	\[ \left(\frac{i}{\beta + 2 \cdot i} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]
      *-commutative [=>] 10.3	\[ \left(\frac{i}{\beta + \color{blue}{i \cdot 2}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \cdot 0.25 \]

   if 3.30000000000000008e106 < beta < 6.60000000000000029e124 or 8.7999999999999999e223 < beta

   1. Initial program 62.8

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

   2. Simplified 53.7

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

      [Start] 62.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 63.9	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 53.7	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 14.7

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106} \lor \neg \left(\beta \leq 6.6 \cdot 10^{+124}\right) \land \beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	14476

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \left(\frac{i}{t_0} \cdot \frac{\beta + i}{t_0}\right) \cdot 0.25\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(\left(1 + 2 \cdot \frac{i}{\beta}\right) + \frac{i}{\beta} \cdot -4\right)\\ \end{array} \]

Alternative 2
Error	11.3
Cost	1612

\[\begin{array}{l} t_0 := \frac{\beta}{i + \alpha}\\ t_1 := \beta + i \cdot 2\\ t_2 := \left(\frac{i}{t_1} \cdot \frac{\beta + i}{t_1}\right) \cdot 0.25\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{i}{\beta \cdot t_0}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{t_0}\\ \end{array} \]

Alternative 3
Error	11.4
Cost	973

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124} \lor \neg \left(\beta \leq 8.8 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4
Error	11.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	11.4
Cost	972

\[\begin{array}{l} t_0 := \frac{\beta}{i + \alpha}\\ \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{i}{\beta \cdot t_0}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{t_0}\\ \end{array} \]

Alternative 6
Error	12.3
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124} \lor \neg \left(\beta \leq 9.2 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 7
Error	12.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	12.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+106}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{i}{\frac{\beta}{\frac{i}{\beta}}}\\ \mathbf{elif}\;\beta \leq 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	18.4
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023083 
(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)))