Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 4.1s

Precision: binary64

Cost: 19776

Math

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

↓

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

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(b, Float64(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[(b * N[(b * 4.0), $MachinePrecision] + -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

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	7424

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

Alternative 2
Error	2.3
Cost	7304

\[\begin{array}{l} t_0 := -1 + {a}^{4}\\ \mathbf{if}\;a \leq -0.09526439621463072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.600598030530978 \cdot 10^{-35}:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.3
Cost	7240

\[\begin{array}{l} t_0 := -1 + {a}^{4}\\ \mathbf{if}\;a \leq -0.09526439621463072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.600598030530978 \cdot 10^{-35}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.3
Cost	6920

\[\begin{array}{l} t_0 := -1 + {a}^{4}\\ \mathbf{if}\;a \leq -0.09526439621463072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.600598030530978 \cdot 10^{-35}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	2.2
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -14.400436092630466:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.01936653645955238:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6
Error	11.4
Cost	704

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

Alternative 7
Error	12.3
Cost	576

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

Alternative 8
Error	22.8
Cost	448

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

Reproduce

herbie shell --seed 2022222 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))