Octave 3.8, jcobi/4

Average Error: 54.0 → 9.2

Time: 21.4s

Precision: binary64

Cost: 14276

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+170}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t_0 + 1}}{\frac{t_0 + -1}{i + \alpha}}\\ \end{array} \]

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))))
   (if (<= beta 4.5e+170)
     (* (* (/ i t_0) (/ (+ i (+ beta alpha)) t_0)) 0.25)
     (/ (/ i (+ t_0 1.0)) (/ (+ t_0 -1.0) (+ i alpha))))))

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 tmp;
	if (beta <= 4.5e+170) {
		tmp = ((i / t_0) * ((i + (beta + alpha)) / t_0)) * 0.25;
	} else {
		tmp = (i / (t_0 + 1.0)) / ((t_0 + -1.0) / (i + alpha));
	}
	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))
	tmp = 0.0
	if (beta <= 4.5e+170)
		tmp = Float64(Float64(Float64(i / t_0) * Float64(Float64(i + Float64(beta + alpha)) / t_0)) * 0.25);
	else
		tmp = Float64(Float64(i / Float64(t_0 + 1.0)) / Float64(Float64(t_0 + -1.0) / Float64(i + alpha)));
	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]}, If[LessEqual[beta, 4.5e+170], N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(i / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.50000000000000022e170

   1. Initial program 50.2

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

   2. Simplified 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] 50.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 51.8	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 33.2	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 7.6

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

   if 4.50000000000000022e170 < 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. Taylor expanded in beta around inf 47.1

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

   3. Applied egg-rr 33.4

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

   4. Simplified 13.5

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

      [Start] 33.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*r/ [=>] 33.4	\[ \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}} \]
      associate-*l/ [=>] 33.4	\[ \frac{\color{blue}{\frac{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-lft-in [=>] 33.4	\[ \frac{\frac{1 \cdot \color{blue}{\left(i \cdot i + i \cdot \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-rgt-out [<=] 33.4	\[ \frac{\frac{\color{blue}{\left(i \cdot i\right) \cdot 1 + \left(i \cdot \alpha\right) \cdot 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      *-rgt-identity [=>] 33.4	\[ \frac{\frac{\color{blue}{i \cdot i} + \left(i \cdot \alpha\right) \cdot 1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      *-rgt-identity [=>] 33.4	\[ \frac{\frac{i \cdot i + \color{blue}{i \cdot \alpha}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      distribute-lft-in [<=] 33.4	\[ \frac{\frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*l/ [<=] 13.5	\[ \frac{\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(i + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-/l* [=>] 13.5	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}}} \]
      +-commutative [=>] 13.5	\[ \frac{\frac{i}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}} \]
      +-commutative [=>] 13.5	\[ \frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{i + \alpha}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+170}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{i + \alpha}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+172}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	7748

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+173}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.3 \cdot 10^{+172}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Error	9.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+172}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 5
Error	10.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+173}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	18.8
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023057 
(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)))