Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 8.0s

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	1.5
Cost	7684

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{-7}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\\ \end{array} \]

Alternative 2
Error	1.6
Cost	7560

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0012:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;a \leq 0.004:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left({a}^{3} + \left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7556

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{-7}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right)\right)\\ \end{array} \]

Alternative 4
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 5
Error	1.6
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0016:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;a \leq 0.00105:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot \left(b \cdot 2\right)\right)\\ \end{array} \]

Alternative 6
Error	1.7
Cost	6788

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0075:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;a \leq 0.0008:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot \left(b \cdot 2\right)\right)\\ \end{array} \]

Alternative 7
Error	1.7
Cost	1220

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{-7}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot \left(b \cdot 2\right)\right)\\ \end{array} \]

Alternative 8
Error	12.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -0.95 \lor \neg \left(b \leq 0.94\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 9
Error	12.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;b \leq -1.95 \lor \neg \left(b \leq 1.92\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 10
Error	11.8
Cost	704

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

Alternative 11
Error	51.7
Cost	448

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

Alternative 12
Error	61.8
Cost	320

\[4 \cdot \left(b \cdot b\right) \]

Reproduce

herbie shell --seed 2023045 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))