Average Error: 0.2 → 0.0
Time: 1.6s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[\left({\left(\sqrt{2} \cdot \left(a \cdot b\right)\right)}^{2} + \left({a}^{4} + {b}^{4}\right)\right) + \mathsf{fma}\left(b, b \cdot 4, -1\right) \]
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (+ (pow (* (sqrt 2.0) (* a b)) 2.0) (+ (pow a 4.0) (pow b 4.0)))
  (fma b (* b 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((sqrt(2.0) * (a * b)), 2.0) + (pow(a, 4.0) + pow(b, 4.0))) + fma(b, (b * 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(Float64((Float64(sqrt(2.0) * Float64(a * b)) ^ 2.0) + Float64((a ^ 4.0) + (b ^ 4.0))) + fma(b, Float64(b * 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[(N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left(\sqrt{2} \cdot \left(a \cdot b\right)\right)}^{2} + \left({a}^{4} + {b}^{4}\right)\right) + \mathsf{fma}\left(b, b \cdot 4, -1\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. Taylor expanded in a around 0 0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022175 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))