Average Error: 0.2 → 0.2
Time: 24.0s
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 r189286 = a;
        double r189287 = r189286 * r189286;
        double r189288 = b;
        double r189289 = r189288 * r189288;
        double r189290 = r189287 + r189289;
        double r189291 = 2.0;
        double r189292 = pow(r189290, r189291);
        double r189293 = 4.0;
        double r189294 = r189293 * r189289;
        double r189295 = r189292 + r189294;
        double r189296 = 1.0;
        double r189297 = r189295 - r189296;
        return r189297;
}
↓
double f(double a, double b) {
        double r189298 = 4.0;
        double r189299 = b;
        double r189300 = r189298 * r189299;
        double r189301 = a;
        double r189302 = r189299 * r189299;
        double r189303 = fma(r189301, r189301, r189302);
        double r189304 = 2.0;
        double r189305 = pow(r189303, r189304);
        double r189306 = fma(r189300, r189299, r189305);
        double r189307 = 1.0;
        double r189308 = r189306 - r189307;
        return r189308;
}

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 2019235 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))