Average Error: 0.2 → 0.2
Time: 4.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left({a}^{2}, \left(1 + a\right) \cdot 4, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(\mathsf{fma}\left({a}^{2}, \left(1 + a\right) \cdot 4, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\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) * (1.0 - (3.0 * a)))))) - 1.0);
}
↓
double code(double a, double b) {
	return ((fma(pow(a, 2.0), ((1.0 + a) * 4.0), pow(((a * a) + (b * b)), 2.0)) + (((b * b) * (1.0 - (3.0 * 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

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

  3. Applied distribute-rgt-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))