Average Error: 0.2 → 0.5
Time: 4.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}, 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}, 4 \cdot \left(b \cdot b\right)\right) - 1
double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (b * b)))))) - 1.0));
}
↓
double code(double a, double b) {
	return ((double) (((double) fma(((double) (((double) cbrt(((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)))) * ((double) cbrt(((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)))))), ((double) cbrt(((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)))), ((double) (4.0 * ((double) (b * b)))))) - 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. Using strategy rm

  3. Applied add-cube-cbrt0.5

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

  4. Applied fma-def0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))