Average Error: 0.2 → 0.2
Time: 5.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right) \cdot \sqrt[3]{4}\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right) \cdot \sqrt[3]{4}\right)\right) - 1
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 ((pow(((a * a) + (b * b)), 2.0) + ((cbrt(4.0) * cbrt(4.0)) * (fma((a * a), (1.0 - a), ((b * b) * (3.0 + a))) * cbrt(4.0)))) - 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

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

  3. Applied add-cube-cbrt0.2

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

  4. Applied associate-*l*0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))