Average Error: 0.2 → 0.2
Time: 20.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r227460 = a;
        double r227461 = r227460 * r227460;
        double r227462 = b;
        double r227463 = r227462 * r227462;
        double r227464 = r227461 + r227463;
        double r227465 = 2.0;
        double r227466 = pow(r227464, r227465);
        double r227467 = 4.0;
        double r227468 = r227467 * r227463;
        double r227469 = r227466 + r227468;
        double r227470 = 1.0;
        double r227471 = r227469 - r227470;
        return r227471;
}
↓
double f(double a, double b) {
        double r227472 = 4.0;
        double r227473 = b;
        double r227474 = r227472 * r227473;
        double r227475 = a;
        double r227476 = r227473 * r227473;
        double r227477 = fma(r227475, r227475, r227476);
        double r227478 = 2.0;
        double r227479 = pow(r227477, r227478);
        double r227480 = fma(r227474, r227473, r227479);
        double r227481 = 1.0;
        double r227482 = r227480 - r227481;
        return r227482;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))