Average Error: 0.2 → 0.0
Time: 3.8s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(4, \left({a}^{2} + \left({a}^{3} + {b}^{2}\right)\right) - 3 \cdot \left(a \cdot {b}^{2}\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (fma
   4.0
   (- (+ (pow a 2.0) (+ (pow a 3.0) (pow b 2.0))) (* 3.0 (* a (pow b 2.0))))
   (pow (hypot a b) 4.0))
  -1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return fma(4.0, ((pow(a, 2.0) + (pow(a, 3.0) + pow(b, 2.0))) - (3.0 * (a * pow(b, 2.0)))), pow(hypot(a, 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(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end
↓
function code(a, b)
	return Float64(fma(4.0, Float64(Float64((a ^ 2.0) + Float64((a ^ 3.0) + (b ^ 2.0))) - Float64(3.0 * Float64(a * (b ^ 2.0)))), (hypot(a, 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[a_, b_] := N[(N[(4.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 * N[(a * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \left({a}^{2} + \left({a}^{3} + {b}^{2}\right)\right) - 3 \cdot \left(a \cdot {b}^{2}\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1} \]

  3. Taylor expanded in a around 0 0.0

    \[\leadsto \mathsf{fma}\left(4, \color{blue}{\left({a}^{2} + \left({a}^{3} + {b}^{2}\right)\right) - 3 \cdot \left(a \cdot {b}^{2}\right)}, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \left({a}^{2} + \left({a}^{3} + {b}^{2}\right)\right) - 3 \cdot \left(a \cdot {b}^{2}\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]

Reproduce

herbie shell --seed 2022146 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))