Average Error: 0.2 → 0.2
Time: 27.9s
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 r254191 = a;
        double r254192 = r254191 * r254191;
        double r254193 = b;
        double r254194 = r254193 * r254193;
        double r254195 = r254192 + r254194;
        double r254196 = 2.0;
        double r254197 = pow(r254195, r254196);
        double r254198 = 4.0;
        double r254199 = r254198 * r254194;
        double r254200 = r254197 + r254199;
        double r254201 = 1.0;
        double r254202 = r254200 - r254201;
        return r254202;
}
↓
double f(double a, double b) {
        double r254203 = 4.0;
        double r254204 = b;
        double r254205 = r254203 * r254204;
        double r254206 = a;
        double r254207 = r254204 * r254204;
        double r254208 = fma(r254206, r254206, r254207);
        double r254209 = 2.0;
        double r254210 = pow(r254208, r254209);
        double r254211 = fma(r254205, r254204, r254210);
        double r254212 = 1.0;
        double r254213 = r254211 - r254212;
        return r254213;
}

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 2019322 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))