Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 6.5s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double a, double b) {
	return pow(hypot(a, b), 4.0) + fma((b * 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(b * b))) - 1.0)
end

↓

function code(a, b)
	return Float64((hypot(a, b) ^ 4.0) + fma(Float64(b * 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

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

   [Start] 0.2	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   associate--l+ [=>] 0.2	\[ \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]
   unpow2 [=>] 0.2	\[ \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   unpow1 [<=] 0.2	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   sqr-pow [=>] 0.2	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   associate-*r* [=>] 0.1	\[ \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   *-commutative [<=] 0.1	\[ \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot \left(a \cdot a + b \cdot b\right)\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	2.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{elif}\;b \leq 1.3:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + 2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7424

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

Alternative 3
Error	1.8
Cost	7300

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-17}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \mathbf{elif}\;b \leq 3.6:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	2.2
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 5.4:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	2.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{elif}\;b \leq 7.8:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 7
Error	2.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 1.92:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 8
Error	1.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 6 \cdot 10^{-15}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 9
Error	2.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \lor \neg \left(b \leq 1.42\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 10
Error	12.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;b \leq -1.95 \lor \neg \left(b \leq 0.00038\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 11
Error	51.7
Cost	448

\[\left(b \cdot b\right) \cdot \left(b \cdot b\right) \]

Reproduce

herbie shell --seed 2023073 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))