Average Error: 0.2 → 0.2
Time: 25.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r156210 = a;
        double r156211 = r156210 * r156210;
        double r156212 = b;
        double r156213 = r156212 * r156212;
        double r156214 = r156211 + r156213;
        double r156215 = 2.0;
        double r156216 = pow(r156214, r156215);
        double r156217 = 4.0;
        double r156218 = 1.0;
        double r156219 = r156218 + r156210;
        double r156220 = r156211 * r156219;
        double r156221 = 3.0;
        double r156222 = r156221 * r156210;
        double r156223 = r156218 - r156222;
        double r156224 = r156213 * r156223;
        double r156225 = r156220 + r156224;
        double r156226 = r156217 * r156225;
        double r156227 = r156216 + r156226;
        double r156228 = r156227 - r156218;
        return r156228;
}
↓
double f(double a, double b) {
        double r156229 = 4.0;
        double r156230 = a;
        double r156231 = r156230 * r156230;
        double r156232 = 1.0;
        double r156233 = r156232 + r156230;
        double r156234 = b;
        double r156235 = r156234 * r156234;
        double r156236 = 3.0;
        double r156237 = r156236 * r156230;
        double r156238 = r156232 - r156237;
        double r156239 = r156235 * r156238;
        double r156240 = fma(r156231, r156233, r156239);
        double r156241 = fma(r156230, r156230, r156235);
        double r156242 = 2.0;
        double r156243 = pow(r156241, r156242);
        double r156244 = fma(r156229, r156240, r156243);
        double r156245 = r156244 - r156232;
        return r156245;
}

Error

Bits error versus a

Bits error versus b

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. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019235 +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))