Average Error: 0.2 → 0.2
Time: 24.8s
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 r225477 = a;
        double r225478 = r225477 * r225477;
        double r225479 = b;
        double r225480 = r225479 * r225479;
        double r225481 = r225478 + r225480;
        double r225482 = 2.0;
        double r225483 = pow(r225481, r225482);
        double r225484 = 4.0;
        double r225485 = r225484 * r225480;
        double r225486 = r225483 + r225485;
        double r225487 = 1.0;
        double r225488 = r225486 - r225487;
        return r225488;
}
↓
double f(double a, double b) {
        double r225489 = 4.0;
        double r225490 = b;
        double r225491 = r225489 * r225490;
        double r225492 = a;
        double r225493 = r225490 * r225490;
        double r225494 = fma(r225492, r225492, r225493);
        double r225495 = 2.0;
        double r225496 = pow(r225494, r225495);
        double r225497 = fma(r225491, r225490, r225496);
        double r225498 = 1.0;
        double r225499 = r225497 - r225498;
        return r225499;
}

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 2019304 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))