Octave 3.8, jcobi/4

Average Error: 53.6 → 0.5

Time: 27.3s

Precision: binary64

Cost: 9408

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{t_0}{i + \left(\alpha + \beta\right)}} \cdot \frac{i + \alpha}{\left(t_0 + -1\right) \cdot \frac{\beta + i \cdot 2}{i + \beta}} \end{array} \]

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 (+ (+ alpha beta) (* i 2.0))))
   (*
    (/ i (* (+ alpha (+ beta (fma i 2.0 1.0))) (/ t_0 (+ i (+ alpha beta)))))
    (/ (+ i alpha) (* (+ t_0 -1.0) (/ (+ beta (* i 2.0)) (+ i 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 = (alpha + beta) + (i * 2.0);
	return (i / ((alpha + (beta + fma(i, 2.0, 1.0))) * (t_0 / (i + (alpha + beta))))) * ((i + alpha) / ((t_0 + -1.0) * ((beta + (i * 2.0)) / (i + beta))));
}

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(alpha + beta) + Float64(i * 2.0))
	return Float64(Float64(i / Float64(Float64(alpha + Float64(beta + fma(i, 2.0, 1.0))) * Float64(t_0 / Float64(i + Float64(alpha + beta))))) * Float64(Float64(i + alpha) / Float64(Float64(t_0 + -1.0) * Float64(Float64(beta + Float64(i * 2.0)) / Float64(i + beta)))))
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[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(i / N[(N[(alpha + N[(beta + N[(i * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.6

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

2. Taylor expanded in beta around 0 53.6

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

3. Applied egg-rr 0.2

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

4. Simplified 0.3

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

   [Start] 0.2	\[ \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \left(\alpha + \beta\right)}}}{\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   associate-/l/ [=>] 0.3	\[ \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + \left(i \cdot 2 + 1\right)\right) \cdot \frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \left(\alpha + \beta\right)}}} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   associate-+l+ [=>] 0.3	\[ \frac{i}{\color{blue}{\left(\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)\right)} \cdot \frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   fma-def [=>] 0.3	\[ \frac{i}{\left(\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(i, 2, 1\right)}\right)\right) \cdot \frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   +-commutative [<=] 0.3	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\color{blue}{\left(\beta + \alpha\right)} + i \cdot 2}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   +-commutative [<=] 0.3	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \color{blue}{\left(\beta + \alpha\right)}}} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]
   associate-/l/ [=>] 0.3	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}} \cdot \color{blue}{\frac{i + \alpha}{\left(\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)\right) \cdot \frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}} \]

5. Taylor expanded in alpha around 0 0.5

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

6. Simplified 0.5

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

   [Start] 0.5	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}} \cdot \frac{i + \alpha}{\left(\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1\right) \cdot \frac{\beta + 2 \cdot i}{\beta + i}} \]
   +-commutative [=>] 0.5	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}} \cdot \frac{i + \alpha}{\left(\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1\right) \cdot \frac{\color{blue}{2 \cdot i + \beta}}{\beta + i}} \]
   +-commutative [=>] 0.5	\[ \frac{i}{\left(\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \left(\beta + \alpha\right)}} \cdot \frac{i + \alpha}{\left(\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1\right) \cdot \frac{2 \cdot i + \beta}{\color{blue}{i + \beta}}} \]

7. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.2
Cost	3776

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

Alternative 2
Error	8.3
Cost	3012

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := \frac{t_0}{i + \beta}\\ \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{i + \alpha}{i \cdot 2 + \left(\alpha + 1\right)} \cdot \frac{i}{\alpha + i \cdot 2}\right) \cdot \frac{i + \alpha}{\left(t_0 + -1\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{t_1}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)} \cdot \frac{i}{\left(\alpha + \beta\right) + \left(1 + i \cdot 2\right)}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	3008

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \frac{\frac{i}{\frac{t_0}{i + \beta}}}{\left(\alpha + \beta\right) + \left(1 + i \cdot 2\right)} \cdot \frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(t_0 + -1\right)} \end{array} \]

Alternative 4
Error	8.5
Cost	2500

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{i}{i \cdot 16 + \frac{-4}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)} \cdot \frac{i}{\left(\alpha + \beta\right) + \left(1 + i \cdot 2\right)}\\ \end{array} \]

Alternative 5
Error	10.3
Cost	2253

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+132}:\\ \;\;\;\;\frac{i}{i \cdot 16 + \frac{-4}{i}}\\ \mathbf{elif}\;\beta \leq 3.35 \cdot 10^{+162} \lor \neg \left(\beta \leq 8.5 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{i + \alpha}{\left(t_0 + -1\right) \cdot \frac{t_0}{i + \beta}} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 6
Error	10.2
Cost	2252

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := \frac{t_0}{i + \beta}\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{i}{i \cdot 16 + \frac{-4}{i}}\\ \mathbf{elif}\;\beta \leq 4 \cdot 10^{+162}:\\ \;\;\;\;\frac{i + \alpha}{\left(t_0 + -1\right) \cdot t_1} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{+212}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\frac{i + \alpha}{\left(\alpha + \beta\right) + \left(i \cdot 2 + -1\right)}}{t_1}\\ \end{array} \]

Alternative 7
Error	10.3
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{i}{i \cdot 16 + \frac{-4}{i}}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 10^{+213}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	9.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	9.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\frac{i}{i \cdot 16 + \frac{-4}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 10
Error	16.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 11
Error	10.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 12
Error	10.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+132}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 13
Error	16.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	57.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023073 
(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)))