Octave 3.8, jcobi/4

Average Error: 54.6 → 9.4

Time: 15.7s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6400779121446645 \cdot 10^{+188}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

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
 (if (<= beta 1.6400779121446645e+188)
   0.0625
   (*
    (/ (+ i alpha) (+ (fma i 2.0 (+ beta alpha)) 1.0))
    (/ i (+ (+ beta alpha) (fma i 2.0 -1.0))))))

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 tmp;
	if (beta <= 1.6400779121446645e+188) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (fma(i, 2.0, (beta + alpha)) + 1.0)) * (i / ((beta + alpha) + fma(i, 2.0, -1.0)));
	}
	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)
	tmp = 0.0
	if (beta <= 1.6400779121446645e+188)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(fma(i, 2.0, Float64(beta + alpha)) + 1.0)) * Float64(i / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))));
	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_] := If[LessEqual[beta, 1.6400779121446645e+188], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6400779121446645e188

   1. Initial program 51.5

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

   2. Taylor expanded in i around inf 8.5

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

   3. Applied egg-rr 8.5

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

   4. Taylor expanded in alpha around 0 8.5

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

   5. Taylor expanded in beta around 0 8.7

      \[\leadsto \color{blue}{0.0625} \]

   if 1.6400779121446645e188 < 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 45.6

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

   3. Applied egg-rr 11.6

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6400779121446645 \cdot 10^{+188}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6400779121446645 \cdot 10^{+188}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 2
Error	15.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1388174992940593 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Error	10.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6400779121446645 \cdot 10^{+188}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 4
Error	10.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6400779121446645 \cdot 10^{+188}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	17.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1388174992940593 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	57.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022300 
(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)))