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

↓

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

\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, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), {b}^{2}\right), {a}^{4} + \left(2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + {b}^{4}\right)\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
   (fma a (fma a a a) (pow b 2.0))
   (+ (pow a 4.0) (+ (* 2.0 (* (pow b 2.0) (pow a 2.0))) (pow 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, fma(a, fma(a, a, a), pow(b, 2.0)), (pow(a, 4.0) + ((2.0 * (pow(b, 2.0) * pow(a, 2.0))) + pow(b, 4.0)))) + -1.0;
}

Derivation

Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1} \]
Taylor expanded in a around 0 0.0
\[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), \color{blue}{{a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {b}^{4}\right)}\right) + -1 \]
Taylor expanded in a around 0 0.1
\[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \color{blue}{{b}^{2}}\right), {a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {b}^{4}\right)\right) + -1 \]
Final simplification0.1
\[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), {b}^{2}\right), {a}^{4} + \left(2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + {b}^{4}\right)\right) + -1 \]

Reproduce

herbie shell --seed 2021332 
(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))

Error

Derivation

Reproduce