Average Error: 0.2 → 0.2
Time: 27.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r9440045 = a;
        double r9440046 = r9440045 * r9440045;
        double r9440047 = b;
        double r9440048 = r9440047 * r9440047;
        double r9440049 = r9440046 + r9440048;
        double r9440050 = 2.0;
        double r9440051 = pow(r9440049, r9440050);
        double r9440052 = 4.0;
        double r9440053 = r9440052 * r9440048;
        double r9440054 = r9440051 + r9440053;
        double r9440055 = 1.0;
        double r9440056 = r9440054 - r9440055;
        return r9440056;
}
↓
double f(double a, double b) {
        double r9440057 = 4.0;
        double r9440058 = b;
        double r9440059 = r9440058 * r9440058;
        double r9440060 = a;
        double r9440061 = fma(r9440060, r9440060, r9440059);
        double r9440062 = 2.0;
        double r9440063 = pow(r9440061, r9440062);
        double r9440064 = fma(r9440057, r9440059, r9440063);
        double r9440065 = 1.0;
        double r9440066 = r9440064 - r9440065;
        return r9440066;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))