Average Error: 0.2 → 0.0
Time: 6.6s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\left(\mathsf{fma}\left(4, a \cdot a - {a}^{3}, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) - 1 \]
\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(3 + a\right)\right)\right) - 1
↓
\left(\mathsf{fma}\left(4, a \cdot a - {a}^{3}, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) - 1

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+
   (fma 4.0 (- (* a a) (pow a 3.0)) (pow (hypot a b) 4.0))
   (* 4.0 (* (* b b) (+ a 3.0))))
  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) * (3.0 + a))))) - 1.0;
}
↓
double code(double a, double b) {
	return (fma(4.0, ((a * a) - pow(a, 3.0)), pow(hypot(a, b), 4.0)) + (4.0 * ((b * b) * (a + 3.0)))) - 1.0;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Applied distribute-lft-in_binary640.2

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

  3. Applied associate-+r+_binary640.2

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022066 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))