Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 9.1s

Precision: binary64

Cost: 20928

Math

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

↓

\[\mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \left({a}^{4} + {b}^{4}\right) + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

↓

(FPCore (a b)
 :precision binary64
 (+
  (fma
   (* b b)
   (+ 12.0 (* 2.0 (* a a)))
   (+ (+ (pow a 4.0) (pow b 4.0)) (* 4.0 (* a (* a (- 1.0 a))))))
  -1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

↓

double code(double a, double b) {
	return fma((b * b), (12.0 + (2.0 * (a * a))), ((pow(a, 4.0) + pow(b, 4.0)) + (4.0 * (a * (a * (1.0 - a)))))) + -1.0;
}

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

↓

function code(a, b)
	return Float64(fma(Float64(b * b), Float64(12.0 + Float64(2.0 * Float64(a * a))), Float64(Float64((a ^ 4.0) + (b ^ 4.0)) + Float64(4.0 * Float64(a * Float64(a * Float64(1.0 - a)))))) + -1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * N[(12.0 + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

2. Taylor expanded in a around 0 0.0

   \[\leadsto \left(\color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

3. Simplified 0.0

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(2, {\left(b \cdot a\right)}^{2}, {b}^{4} + {a}^{4}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   Proof

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   fma-def [=>] 0.0	\[ \left(\color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {a}^{4} + {b}^{4}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}, {a}^{4} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}, {a}^{4} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   swap-sqr [<=] 0.0	\[ \left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}, {a}^{4} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [<=] 0.0	\[ \left(\mathsf{fma}\left(2, \color{blue}{{\left(a \cdot b\right)}^{2}}, {a}^{4} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   *-commutative [=>] 0.0	\[ \left(\mathsf{fma}\left(2, {\color{blue}{\left(b \cdot a\right)}}^{2}, {a}^{4} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   +-commutative [=>] 0.0	\[ \left(\mathsf{fma}\left(2, {\left(b \cdot a\right)}^{2}, \color{blue}{{b}^{4} + {a}^{4}}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

4. Applied egg-rr 0.0

   \[\leadsto \left(\mathsf{fma}\left(2, \color{blue}{\left(\left(b \cdot a\right) \cdot a\right) \cdot b}, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

5. Taylor expanded in a around 0 0.1

   \[\leadsto \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{3 \cdot {b}^{2}}\right)\right) - 1 \]

6. Simplified 0.1

   \[\leadsto \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{b \cdot \left(b \cdot 3\right)}\right)\right) - 1 \]
   Proof

   [Start] 0.1	\[ \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + 3 \cdot {b}^{2}\right)\right) - 1 \]
   *-commutative [=>] 0.1	\[ \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{{b}^{2} \cdot 3}\right)\right) - 1 \]
   unpow2 [=>] 0.1	\[ \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\left(b \cdot b\right)} \cdot 3\right)\right) - 1 \]
   associate-*r* [<=] 0.1	\[ \left(\mathsf{fma}\left(2, \left(\left(b \cdot a\right) \cdot a\right) \cdot b, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{b \cdot \left(b \cdot 3\right)}\right)\right) - 1 \]

7. Taylor expanded in b around 0 0.1

   \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(12 + 2 \cdot {a}^{2}\right) + \left({b}^{4} + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)\right)} - 1 \]

8. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \left({a}^{4} + {b}^{4}\right) + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]
   Proof

   [Start] 0.1	\[ \left({b}^{2} \cdot \left(12 + 2 \cdot {a}^{2}\right) + \left({b}^{4} + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)\right) - 1 \]
   associate-+r+ [=>] 0.1	\[ \left({b}^{2} \cdot \left(12 + 2 \cdot {a}^{2}\right) + \color{blue}{\left(\left({b}^{4} + {a}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)}\right) - 1 \]
   +-commutative [<=] 0.1	\[ \left({b}^{2} \cdot \left(12 + 2 \cdot {a}^{2}\right) + \left(\color{blue}{\left({a}^{4} + {b}^{4}\right)} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) - 1 \]
   associate-+r+ [<=] 0.1	\[ \left({b}^{2} \cdot \left(12 + 2 \cdot {a}^{2}\right) + \color{blue}{\left({a}^{4} + \left({b}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)}\right) - 1 \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left({b}^{2}, 12 + 2 \cdot {a}^{2}, {a}^{4} + \left({b}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right)} - 1 \]
   unpow2 [=>] 0.1	\[ \mathsf{fma}\left(\color{blue}{b \cdot b}, 12 + 2 \cdot {a}^{2}, {a}^{4} + \left({b}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) - 1 \]
   unpow2 [=>] 0.1	\[ \mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \color{blue}{\left(a \cdot a\right)}, {a}^{4} + \left({b}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) - 1 \]
   associate-+r+ [=>] 0.1	\[ \mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \color{blue}{\left({a}^{4} + {b}^{4}\right) + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)}\right) - 1 \]
   unpow2 [=>] 0.1	\[ \mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \left({a}^{4} + {b}^{4}\right) + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right) - 1 \]
   associate-*r* [<=] 0.1	\[ \mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \left({a}^{4} + {b}^{4}\right) + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)}\right) - 1 \]

9. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(b \cdot b, 12 + 2 \cdot \left(a \cdot a\right), \left({a}^{4} + {b}^{4}\right) + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

Alternatives

Alternative 1
Error	0.2
Cost	8068

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(a \cdot a + b \cdot \left(b \cdot 3\right)\right)\right) + -1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	8064

\[\left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot 3\right)\right)\right) + -1 \]

Alternative 3
Error	1.5
Cost	7561

\[\begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-8} \lor \neg \left(b \leq 8.5 \cdot 10^{-11}\right):\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	1.7
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 5
Error	2.5
Cost	7433

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-8} \lor \neg \left(b \leq 8.5 \cdot 10^{-11}\right):\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + -4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	2.5
Cost	7305

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-8} \lor \neg \left(b \leq 8.5 \cdot 10^{-11}\right):\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \end{array} \]

Alternative 7
Error	2.6
Cost	7241

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-9} \lor \neg \left(b \leq 8.5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \end{array} \]

Alternative 8
Error	3.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -0.34:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 1.15:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9
Error	3.3
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -0.38:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 9.2:\\ \;\;\;\;\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 10
Error	13.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \leq -4.8 \lor \neg \left(a \leq 2.45\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 \cdot \left(1 - a\right)\right)\right) + -1\\ \end{array} \]

Alternative 11
Error	13.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -0.42 \lor \neg \left(a \leq 2.45\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12
Error	13.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -1.95 \lor \neg \left(a \leq 1.1\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \]

Alternative 13
Error	12.8
Cost	704

\[\left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1 \]

Alternative 14
Error	23.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023054 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))