Average Error: 0.2 → 0.2
Time: 6.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, 3 \cdot \left(b \cdot b\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot 4\right) - 1\]
\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, \mathsf{fma}\left(a \cdot a, 1 - a, 3 \cdot \left(b \cdot b\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot 4\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) (((double) (((double) (a * a)) * ((double) (1.0 - a)))) + ((double) (((double) (b * b)) * ((double) (3.0 + a)))))))))) - 1.0));
}
↓
double code(double a, double b) {
	return ((double) (((double) (((double) fma(4.0, ((double) fma(((double) (a * a)), ((double) (1.0 - a)), ((double) (3.0 * ((double) (b * b)))))), ((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)))) + ((double) (((double) (((double) (b * b)) * a)) * 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 distribute-lft-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Applied distribute-rgt-in0.2

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

  6. Applied associate-+r+0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020114 +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))