Octave 3.8, jcobi/4

Average Error: 53.5 → 2.0

Time: 21.8s

Precision: binary64

Cost: 27712

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

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

Math

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

↓

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

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

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

Derivation

1. Initial program 53.5

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

2. Taylor expanded in alpha around 0 53.6

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

3. Simplified 41.5

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

   [Start] 53.6	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 41.5	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 41.5	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [<=] 41.5	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 2.0

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

5. Taylor expanded in alpha around 0 39.4

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

6. Simplified 2.0

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

   [Start] 39.4	\[ \frac{i \cdot \left(\beta + i\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [<=] 39.4	\[ \frac{i \cdot \color{blue}{\left(i + \beta\right)}}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   times-frac [=>] 2.0	\[ \color{blue}{\left(\frac{i}{\beta + 2 \cdot i} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [=>] 2.0	\[ \left(\frac{i}{\color{blue}{2 \cdot i + \beta}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   *-commutative [=>] 2.0	\[ \left(\frac{i}{\color{blue}{i \cdot 2} + \beta} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   fma-udef [<=] 2.0	\[ \left(\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{\left(1 + 2 \cdot i\right) + \beta}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   associate-+l+ [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{1 + \left(2 \cdot i + \beta\right)}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   *-commutative [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \left(\color{blue}{i \cdot 2} + \beta\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   fma-udef [<=] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]

7. Applied egg-rr 2.0

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

8. Simplified 2.0

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

   [Start] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \left(\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}}{1} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}\right) \]
   /-rgt-identity [=>] 2.0	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \left(\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}\right) \]

9. Final simplification 2.0

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

Alternatives

Alternative 1
Error	2.1
Cost	27712

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

Alternative 2
Error	9.9
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Error	9.9
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + \beta \cdot \frac{0.125}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 6
Error	9.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+184}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	10.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 8
Error	16.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	15.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 11
Error	16.8
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 12
Error	18.8
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023039 
(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)))