Average Error: 0.2 → 0.0
Time: 3.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} - 1\right) + \left(1 - 1\right)\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4 \cdot b, b, \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} - 1\right) + \left(1 - 1\right)\right)
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((4.0 * b), b, ((pow(hypot(a, b), (2.0 * 2.0)) - 1.0) + (1.0 - 1.0)));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied add-sqr-sqrt0.2

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

  6. Applied unpow-prod-down0.2

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

  7. Applied prod-diff0.2

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))