Average Error: 0.2 → 0.0
Time: 2.6s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(b, b \cdot 4, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(b, b \cdot 4, {\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 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+ (fma b (* b 4.0) (pow (hypot a b) 4.0)) -1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}
↓
double code(double a, double b) {
	return fma(b, (b * 4.0), pow(hypot(a, 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. Applied add-cbrt-cube_binary6415.7

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

  4. Simplified15.7

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

  5. Applied fma-udef_binary6415.7

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

  6. Applied associate-+r+_binary6415.6

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2022095 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))