Average Error: 0.2 → 0.0
Time: 2.5s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(b, b \cdot 4, -1 + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) \]
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (fma b (* b 4.0) (+ -1.0 (pow (hypot a b) 4.0))))
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 fma(b, (b * 4.0), (-1.0 + pow(hypot(a, b), 4.0)));
}

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

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

Error

Bits error versus a

Bits error versus b

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. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022178 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))