Average Error: 0.2 → 0.0
Time: 5.5s
Precision: binary64
Cost: 26560
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[{b}^{4} + \left({a}^{4} + \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(2, a \cdot a, 4\right), -1\right)\right) \]
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+ (pow b 4.0) (+ (pow a 4.0) (fma b (* b (fma 2.0 (* a a) 4.0)) -1.0))))
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(b, 4.0) + (pow(a, 4.0) + fma(b, (b * fma(2.0, (a * a), 4.0)), -1.0));
}

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((b ^ 4.0) + Float64((a ^ 4.0) + fma(b, Float64(b * fma(2.0, Float64(a * a), 4.0)), -1.0)))
end

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[b, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(b * N[(b * N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

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. Simplified0.0

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

  3. Taylor expanded in a around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost19776
\[{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b, b \cdot 4, -1\right) \]
Alternative 2
Error0.2
Cost7424
\[-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) \]
Alternative 3
Error1.5
Cost7300
\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
Alternative 4
Error1.6
Cost7236
\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]
Alternative 5
Error1.6
Cost6916
\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]
Alternative 6
Error2.2
Cost6660
\[\begin{array}{l} \mathbf{if}\;a \leq -460522.1364421929:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 17265.122163642864:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot a + 2 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
Alternative 7
Error2.1
Cost1096
\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a + 2 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{if}\;a \leq -460522.1364421929:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 17265.122163642864:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error2.2
Cost968
\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -460522.1364421929:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 17265.122163642864:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error3.2
Cost840
\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -460522.1364421929:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 17265.122163642864:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error12.1
Cost712
\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -460522.1364421929:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.787956084833792 \cdot 10^{-5}:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error51.1
Cost448
\[a \cdot \left(a \cdot \left(a \cdot a\right)\right) \]

Error

Reproduce

herbie shell --seed 2022225 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))