Octave 3.8, jcobi/4

Average Error: 15.0% → 86.4%

Time: 29.3s

Precision: binary64

Cost: 14276.00

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + t_0}{i + \alpha}}}{1 + t_0}\\ \end{array} \]

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 (+ beta alpha))))
   (if (<= beta 2.5e+157)
     (+ (+ 0.0625 (* 0.125 (/ beta i))) (* (/ beta i) -0.125))
     (/ (/ i (/ (+ -1.0 t_0) (+ i alpha))) (+ 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, (beta + alpha));
	double tmp;
	if (beta <= 2.5e+157) {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	} else {
		tmp = (i / ((-1.0 + t_0) / (i + alpha))) / (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(beta + alpha))
	tmp = 0.0
	if (beta <= 2.5e+157)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	else
		tmp = Float64(Float64(i / Float64(Float64(-1.0 + t_0) / Float64(i + alpha))) / 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+157], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(N[(-1.0 + t$95$0), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.49999999999999988e157

   1. Initial program 21.6

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

   2. Simplified 49.1

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

      [Start] 21.6	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 19.5	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 49.1	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 90.4

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   4. Taylor expanded in beta around inf 90.4

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(2 \cdot \frac{\beta}{i}\right)}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   5. Simplified 90.4

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      Proof

      [Start] 90.4	\[ \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      associate-*r/ [=>] 90.4	\[ \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   6. Taylor expanded in i around 0 90.4

      \[\leadsto \color{blue}{\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   7. Taylor expanded in beta around inf 90.4

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]

   if 2.49999999999999988e157 < beta

   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. Taylor expanded in beta around inf 23.5

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

   3. Applied egg-rr 48.4

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

      [Start] 23.5	\[ \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-un-lft-identity [=>] 23.5	\[ \frac{\color{blue}{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 23.5	\[ \frac{1 \cdot \left(i \cdot \left(i + \alpha\right)\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 48.4	\[ \color{blue}{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]
      +-commutative [=>] 48.4	\[ \frac{1}{\color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)} + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 48.4	\[ \frac{1}{\left(\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 48.4	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      sub-neg [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \left(-1\right)}} \]
      +-commutative [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\left(2 \cdot i + \left(\alpha + \beta\right)\right)} + \left(-1\right)} \]
      *-commutative [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\left(\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)\right) + \left(-1\right)} \]
      fma-def [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} + \left(-1\right)} \]
      metadata-eval [=>] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + \color{blue}{-1}} \]

   4. Simplified 77.2

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

      [Start] 48.4	\[ \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*l/ [=>] 48.4	\[ \color{blue}{\frac{1 \cdot \frac{i \cdot \left(i + \alpha\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}} \]
      associate-/l* [=>] 77.2	\[ \frac{1 \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{i + \alpha}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 77.2	\[ \frac{1 \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{i + \alpha}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \]
      +-commutative [=>] 77.2	\[ \frac{1 \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \alpha}}}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 86.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \alpha}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	86.4%
Cost	14276.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+157}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{1 + t_0} \cdot \frac{i}{-1 + t_0}\\ \end{array} \]

Alternative 2
Error	85.7%
Cost	7364.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	85.6%
Cost	964.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+157}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta + \alpha}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 4
Error	85.6%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta + \alpha}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 5
Error	85.6%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	83.1%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+194}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	74.3%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+245}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	10.0%
Cost	64.00

\[0 \]

Reproduce

herbie shell --seed 2023121 
(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)))