Average Error: 0.2 → 0.0
Time: 3.0s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[{a}^{4} + \left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4\right), {b}^{4}\right) + -1\right) \]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
{a}^{4} + \left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4\right), {b}^{4}\right) + -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 a 4.0) (+ (fma (* b b) (fma 2.0 (* a a) 4.0) (pow 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(a, 4.0) + (fma((b * b), fma(2.0, (a * a), 4.0), pow(b, 4.0)) + -1.0);
}

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

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))