Bouland and Aaronson, Equation (24)

Average Error: 0.29% → 0.05%

Time: 11.9s

Precision: binary64

Cost: 33152

Math

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

↓

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

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
  4.0
  (- (fma (* b b) (+ a 3.0) (* a a)) (pow a 3.0))
  (+ (pow (hypot a b) 4.0) -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(4.0, (fma((b * b), (a + 3.0), (a * a)) - pow(a, 3.0)), (pow(hypot(a, b), 4.0) + -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 fma(4.0, Float64(fma(Float64(b * b), Float64(a + 3.0), Float64(a * a)) - (a ^ 3.0)), Float64((hypot(a, b) ^ 4.0) + -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[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.29

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

2. Simplified 0.05

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

   [Start] 0.29	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   associate--l+ [=>] 0.3	\[ \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)} \]
   +-commutative [=>] 0.3	\[ \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) + {\left(a \cdot a + b \cdot b\right)}^{2}} \]
   associate-+l- [=>] 0.3	\[ \color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - \left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)} \]
   fma-neg [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right), -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)} \]
   +-commutative [=>] 0.3	\[ \mathsf{fma}\left(4, \color{blue}{\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)}, -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   distribute-rgt-out-- [<=] 0.3	\[ \mathsf{fma}\left(4, \left(b \cdot b\right) \cdot \left(3 + a\right) + \color{blue}{\left(1 \cdot \left(a \cdot a\right) - a \cdot \left(a \cdot a\right)\right)}, -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   *-lft-identity [=>] 0.3	\[ \mathsf{fma}\left(4, \left(b \cdot b\right) \cdot \left(3 + a\right) + \left(\color{blue}{a \cdot a} - a \cdot \left(a \cdot a\right)\right), -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   associate-+r- [=>] 0.3	\[ \mathsf{fma}\left(4, \color{blue}{\left(\left(b \cdot b\right) \cdot \left(3 + a\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)}, -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   fma-def [=>] 0.3	\[ \mathsf{fma}\left(4, \color{blue}{\mathsf{fma}\left(b \cdot b, 3 + a, a \cdot a\right)} - a \cdot \left(a \cdot a\right), -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   +-commutative [=>] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, \color{blue}{a + 3}, a \cdot a\right) - a \cdot \left(a \cdot a\right), -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   cube-unmult [=>] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - \color{blue}{{a}^{3}}, -\left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]
   neg-sub0 [=>] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, \color{blue}{0 - \left(1 - {\left(a \cdot a + b \cdot b\right)}^{2}\right)}\right) \]
   associate--r- [=>] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, \color{blue}{\left(0 - 1\right) + {\left(a \cdot a + b \cdot b\right)}^{2}}\right) \]
   metadata-eval [=>] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, \color{blue}{-1} + {\left(a \cdot a + b \cdot b\right)}^{2}\right) \]
   metadata-eval [<=] 0.3	\[ \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, \color{blue}{\left(-1\right)} + {\left(a \cdot a + b \cdot b\right)}^{2}\right) \]

3. Final simplification 0.05

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

Alternatives

Alternative 1
Error	0.17%
Cost	20864

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

Alternative 2
Error	0.3%
Cost	20736

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

Alternative 3
Error	0.31%
Cost	8196

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	0.29%
Cost	8192

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

Alternative 5
Error	2.45%
Cost	7560

\[\begin{array}{l} t_0 := -1 + {b}^{4}\\ \mathbf{if}\;b \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 \cdot \left(a + 3\right)\right) + t_0\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;-1 + a \cdot \left({a}^{3} + \left(1 - a\right) \cdot \left(4 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 6
Error	2.37%
Cost	7560

\[\begin{array}{l} t_0 := -1 + {b}^{4}\\ \mathbf{if}\;b \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 \cdot \left(a + 3\right)\right) + t_0\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 7
Error	3.78%
Cost	7432

\[\begin{array}{l} t_0 := -1 + {b}^{4}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 \cdot \left(a + 3\right)\right) + t_0\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left({a}^{4} + -4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 8
Error	3.79%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + \left({a}^{4} + -4 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + {b}^{4}\right) + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 9
Error	3.83%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + {b}^{4}\right) + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 10
Error	3.92%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]

Alternative 11
Error	3.96%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 12
Error	4.57%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 13
Error	5.11%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;a \leq -7.1 \cdot 10^{+22} \lor \neg \left(a \leq 2.7\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 14
Error	5.21%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \leq -7.1 \cdot 10^{+22} \lor \neg \left(a \leq 3.7\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 15
Error	6.76%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -7.1 \cdot 10^{+22} \lor \neg \left(a \leq 3.8\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 16
Error	20.72%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -0.42 \lor \neg \left(a \leq 0.001\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 17
Error	19.19%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -0.42 \lor \neg \left(a \leq 2.4\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 18
Error	36.32%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023089 
(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))